february 1, 2021 · 2021. 2. 2. · for a non-zero fractional ideal i as in (1) above, the order of...
TRANSCRIPT
![Page 1: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/1.jpg)
Northern Illinois University, Math 681
February 1, 2021
Math 681, Monday, February 1 February 1, 2021
![Page 2: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/2.jpg)
More on Fractional Ideals
As has been our practice so far, K will be a number field with ring ofintegers OK . The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 3: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/3.jpg)
More on Fractional Ideals
As has been our practice so far,
K will be a number field with ring ofintegers OK . The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 4: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/4.jpg)
More on Fractional Ideals
As has been our practice so far, K will be a number field with ring ofintegers OK .
The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 5: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/5.jpg)
More on Fractional Ideals
As has been our practice so far, K will be a number field with ring ofintegers OK . The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 6: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/6.jpg)
More on Fractional Ideals
As has been our practice so far, K will be a number field with ring ofintegers OK . The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 7: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/7.jpg)
More on Fractional Ideals
As has been our practice so far, K will be a number field with ring ofintegers OK . The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 8: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/8.jpg)
More on Fractional Ideals
As has been our practice so far, K will be a number field with ring ofintegers OK . The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 9: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/9.jpg)
More on Fractional Ideals
As has been our practice so far, K will be a number field with ring ofintegers OK . The upper case script German (“fraktur”) font will be usedto denote fractional ideals and the lower case Greek font will be used todenote elements of K .
Recall our big result.
Theorem (Fundamental Theorem)
The set of non-zero fractional ideals of K is a free abelian group on(generated by) the maximal ideals of OK .
In particular, any non-zero ideal I can be expressed uniquely as a productof non-zero prime ideals:
I = Pe11 · · ·P
err . (1)
Math 681, Monday, February 1 February 1, 2021
![Page 10: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/10.jpg)
For two non-zero ideals A and B
we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 11: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/11.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem.
Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 12: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/12.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 13: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/13.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above,
the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 14: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/14.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r .
For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 15: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/15.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P,
theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 16: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/16.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0.
The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 17: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/17.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.
We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 18: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/18.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 19: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/19.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 20: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/20.jpg)
For two non-zero ideals A and B we are on firm ground saying A|B ifB = AC for some non-zero ideal C by the Fundamental Theorem. Notethat A|B if and only if A ⊇ B as sets.
Definition
For a non-zero fractional ideal I as in (1) above, the order of I at themaximal ideal Pi is ei for i = 1, . . . , r . For all other maximal ideals P, theorder of I at P is 0. The order of OK at P is 0 for all maximal ideals P.We write ordP(I) for the order of I at P.
By the Fundamental Theorem, any ideal I is completely determined by theset of numbers
{ordP(I) : P a maximal ideal of OK}.
Math 681, Monday, February 1 February 1, 2021
![Page 21: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/21.jpg)
Definition
Given two non-zero ideals A and B,
we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
![Page 22: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/22.jpg)
Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B)
and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
![Page 23: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/23.jpg)
Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
![Page 24: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/24.jpg)
Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
![Page 25: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/25.jpg)
Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
![Page 26: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/26.jpg)
Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P.
We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
![Page 27: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/27.jpg)
Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
![Page 28: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/28.jpg)
Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals.
For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
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Definition
Given two non-zero ideals A and B, we define the greatest common divisorgcd(A,B) and least common multiple lcm(A,B) of A and B as follows:
ordP
(gcd(A,B)
)= min{ordP(A), ordP(B)},
ordP
(lcm(A,B)
)= max{ordP(A), ordP(B)}
for all maximal ideals P. We say A and B are relatively prime if theirgreatest common divisor is OK .
We will abuse notation just a bit here and extend all of the above toindividual non-zero elements α ∈ K via principal ideals. For example,ordP(α) = ordP
((α)), where (α) is the principal ideal generated by α.
Math 681, Monday, February 1 February 1, 2021
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Note that the gcd(A,B) is the smallest (set-theoretically) ideal whichcontains both A and B.
In other words,
gcd(A,B) = A + B := {α + β : α ∈ A, β ∈ B}.
Similarly, the lcm(A,B) is the largest (set-theoretically) ideal which iscontained in both A and B. It isn’t difficult to see thatgcd(A,B) lcm(A,B) = AB.
Math 681, Monday, February 1 February 1, 2021
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Note that the gcd(A,B) is the smallest (set-theoretically) ideal whichcontains both A and B. In other words,
gcd(A,B) = A + B := {α + β : α ∈ A, β ∈ B}.
Similarly, the lcm(A,B) is the largest (set-theoretically) ideal which iscontained in both A and B. It isn’t difficult to see thatgcd(A,B) lcm(A,B) = AB.
Math 681, Monday, February 1 February 1, 2021
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Note that the gcd(A,B) is the smallest (set-theoretically) ideal whichcontains both A and B. In other words,
gcd(A,B) = A + B := {α + β : α ∈ A, β ∈ B}.
Similarly, the lcm(A,B) is the largest (set-theoretically) ideal which iscontained in both A and B.
It isn’t difficult to see thatgcd(A,B) lcm(A,B) = AB.
Math 681, Monday, February 1 February 1, 2021
![Page 33: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/33.jpg)
Note that the gcd(A,B) is the smallest (set-theoretically) ideal whichcontains both A and B. In other words,
gcd(A,B) = A + B := {α + β : α ∈ A, β ∈ B}.
Similarly, the lcm(A,B) is the largest (set-theoretically) ideal which iscontained in both A and B. It isn’t difficult to see thatgcd(A,B) lcm(A,B) = AB.
Math 681, Monday, February 1 February 1, 2021
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Clearly ordP(AB) = ordP(A) + ordP(B).
Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 35: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/35.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}.
However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 36: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/36.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK .
Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 37: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/37.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β),
we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 38: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/38.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 39: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/39.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 40: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/40.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 41: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/41.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}.
Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 42: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/42.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 43: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/43.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof:
By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 44: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/44.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1,
and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 45: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/45.jpg)
Clearly ordP(AB) = ordP(A) + ordP(B). Since A + B = gcd(A,B), wehave ordP(A + B) = min{ordP(A), ordP(B)}. However, it is not generallythe case that (α) + (β) = (α + β) for α, β ∈ OK . Since(α) + (β)|(α + β), we do have
ordP(α + β) ≥ min{ordP(α), ordP(β)}.
One can check that this is an equality whenever ordP(α) 6= ordP(β).
Lemma (1)
Let A be a non-zero ideal and α ∈ OK \ {0}. Then there is a non-zeroideal B with AB = (α) if and only if α ∈ A.
Proof: By the Fundamental Theorem AB = (α) if and only ifB = (α)A−1, and (α)A−1 ⊆ OK if and only if (α) ⊆ A.
Math 681, Monday, February 1 February 1, 2021
![Page 46: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/46.jpg)
Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals.
Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof:
This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK
(just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1),
so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .
Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB.
To ease notation here, let ei = ordPi(A) for i = 1, . . . , r .
DefineAi = AP1 · · ·PrP
−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .
DefineAi = AP1 · · ·PrP
−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
![Page 55: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/55.jpg)
Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
![Page 56: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/56.jpg)
Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
![Page 57: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/57.jpg)
Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK ,
which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Lemma (2)
Let A and B be non-zero ideals. Then there is an α ∈ A withgcd
((α),AB
)= A.
Proof: This is obvious if A = OK (just use α = 1), so assume A 6= OK .Let P1, . . . ,Pr be the maximal ideals occurring in the unique factorizationof AB. To ease notation here, let ei = ordPi
(A) for i = 1, . . . , r .Define
Ai = AP1 · · ·PrP−ei−1i , i = 1, . . . , r .
Note that
ordPj(Ai ) =
{0 if i = j ,
ej + 1 ≥ 1 otherwise.
Thus, gcd(A1, . . . ,Ar ) = OK , which implies that there are αi ∈ Ai fori = 1, . . . , r with
α1 + · · ·+ αr = 1. (2)
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P,
(2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r
and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r .
This togetherwith (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr ,
we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
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Since each αi ∈ Ai we have
ordPj(αi ) ≥ ordPj
(Ai ) = ej + 1 ≥ 1 i 6= j . (3)
Since ordP(1) = 0 for all maximal ideals P, (2) and (3) imply that
ordPi(αi ) = 0, i = 1, . . . , r . (4)
Now chose βi ∈ Peii \P
ei+1i for all i = 1, . . . , r and let
α = α1β1 + · · ·+ αrβr .
By construction we have ordPi(βi ) = ei for all i = 1, . . . , r . This together
with (3) and (4) show that
ordPi(α) = ei , i = 1, . . . , r .
Since ordP(AB) = 0 for all P not among P1, . . . ,Pr , we havegcd
((α),AB
)= A.
Math 681, Monday, February 1 February 1, 2021
![Page 68: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/68.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
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Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 70: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/70.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}.
Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 71: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/71.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A.
In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 72: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/72.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 73: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/73.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.
Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 74: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/74.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK ,
we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 75: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/75.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A.
We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 76: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/76.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A.
A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 77: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/77.jpg)
Combining Lemmas 1 and 2 gives us the following result.
Lemma (3)
Let A be a non-zero ideal and let β ∈ A \ {0}. Then there is an α ∈ Awith gcd(α, β) = A. In particular, all non-zero ideals can be viewed as thegreatest common divisor of two algebraic integers.
We can speak of congruences in OK in much the same way we do in Z.Specifically, for a non-zero ideal A and α, β ∈ OK , we say α is congruentto β modulo A if α− β ∈ A. We denote this more compactly by writingα ≡ β mod A. A more “advanced” way to say this is α + A = β + A aselements of the quotient ring OK/A.
Math 681, Monday, February 1 February 1, 2021
![Page 78: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/78.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK . Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof: This congruence has a solution if and only if β ∈ A + (α), that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 79: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/79.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK . Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof: This congruence has a solution if and only if β ∈ A + (α), that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 80: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/80.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK .
Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof: This congruence has a solution if and only if β ∈ A + (α), that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 81: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/81.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK . Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof: This congruence has a solution if and only if β ∈ A + (α), that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 82: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/82.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK . Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof: This congruence has a solution if and only if β ∈ A + (α), that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 83: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/83.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK . Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof:
This congruence has a solution if and only if β ∈ A + (α), that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 84: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/84.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK . Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof: This congruence has a solution if and only if β ∈ A + (α),
that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 85: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/85.jpg)
The existence of solutions to linear congruences is very much the same asit is with Z.
Lemma (4)
Let A be a non-zero ideal and let α, β ∈ OK . Then the congruence
Xα ≡ β mod A
has a solution in OK if and only if gcd((α),A
)|(β).
Proof: This congruence has a solution if and only if β ∈ A + (α), that is,(β) ⊆ gcd
((α),A
).
Math 681, Monday, February 1 February 1, 2021
![Page 86: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/86.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 87: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/87.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 88: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/88.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime,
i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 89: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/89.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j .
Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 90: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/90.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar .
Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 91: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/91.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 92: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/92.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK
there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 93: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/93.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 94: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/94.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 95: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/95.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof:
We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 96: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/96.jpg)
We also know when we can solve simultaneous congruences.
Theorem (Chinese Remainder Theorem)
Let A1, . . . ,Ar be non-zero ideals which are pair-wise relatively prime, i.e.,Ai + Aj = OK whenever i 6= j . Let I denote the product A1 · · ·Ar . Then
OK/I ∼= OK/A1 × · · · ×OK/Ar .
In particular, given β1, . . . , βr ∈ OK there is an α ∈ OK with
α ≡ βi mod Ai , i = 1, . . . , r
and this α is unique modulo I.
Proof: We prove this by induction on r . First assume r = 2 and write1 = α1 + α2 with α1 ∈ A1 and α2 ∈ A2.
Math 681, Monday, February 1 February 1, 2021
![Page 97: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/97.jpg)
We readily see that the map
β + I 7→(β + A1, β + A2
)
gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
![Page 98: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/98.jpg)
We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2.
To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
![Page 99: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/99.jpg)
We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK .
Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
![Page 100: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/100.jpg)
We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2)
since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
![Page 101: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/101.jpg)
We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
![Page 102: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/102.jpg)
We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
![Page 103: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/103.jpg)
We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 .
Then gcd(B,A1) = 1 and by the inductionhypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
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We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1
and by the inductionhypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
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We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
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We readily see that the map
β + I 7→(β + A1, β + A2
)gives a well-defined one-to-one ring homomorphism from OK/I toOK/A1 ×OK/A2. To see that it is onto, let γ1, γ2 ∈ OK . Thenγ1α2 + γ2α1 + I is mapped to (γ1 + A1, γ2 + A2) since
α2 ≡ 1 mod A1, α1 ≡ 0 mod A1
α1 ≡ 1 mod A2, α2 ≡ 0 mod A2.
For r > 2, let B = IA−11 . Then gcd(B,A1) = 1 and by the induction
hypothesis (twice) we have
OK/I ∼= OK/A1 ×OK/B ∼= OK/A1 ×OK/A2 × · · · ×OK/Ar .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I],
which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring,
we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals.
Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 113: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/113.jpg)
Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer.
Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 116: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/116.jpg)
Since the norm of a non-zero ideal I is the index [OK : I], which is simplythe cardinality of the quotient ring, we get the following.
Corollary
Let A1, . . . ,Ar be pair-wise relatively prime non-zero ideals. Then
N(A1 · · ·Ar ) = N(A1) · · ·N(Ar ).
Lemma (5)
Let P be a maximal ideal and e be a non-negative integer. Then
[Pe : Pe+1] = N(P).
Thus,N(Pe) = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Proof:
Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
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Proof: Let α ∈ Pe \Pe+1.
Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 119: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/119.jpg)
Proof: Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe .
By Lemma 4,for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 120: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/120.jpg)
Proof: Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.
Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 121: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/121.jpg)
Proof: Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α),
whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 122: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/122.jpg)
Proof: Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2).
In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 123: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/123.jpg)
Proof: Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P.
Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 124: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/124.jpg)
Proof: Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.
Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 125: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/125.jpg)
Proof: Let α ∈ Pe \Pe+1. Then gcd((α),Pe+1
)= Pe . By Lemma 4,
for any β ∈ Pe we can solve the congruence Xα ≡ β mod Pe+1.Moreover, γ1α ≡ γ2α mod Pe+1 if and only if Pe+1|(γ1 − γ2)(α), whichit true if and only if P|(γ1 − γ2). In other words, the solutions to thecongruence Xα ≡ β mod Pe+1 are all congruent modulo P. Thus, thereare precisely N(P) elements of Pe which are incongruent modulo Pe+1.Finally, we have
[OK : Pe ] = [OK : P][P : P2] · · · [Pe−1 : Pe ] = N(P)e .
Math 681, Monday, February 1 February 1, 2021
![Page 126: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/126.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 127: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/127.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 128: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/128.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er
wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
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Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 130: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/130.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 131: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/131.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1).
With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 132: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/132.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers.
Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 133: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/133.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings.
See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 134: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/134.jpg)
Combining the Corollary to the Chinese Remainder Theorem with Lemma5 gives the following.
Theorem
For any maximal ideals P1, . . . ,Pr and non-negative integers e1, . . . , er wehave
N(Pe11 · · ·P
err ) = N(P1)e1 · · ·N(Pr )er .
Given this, it is natural to extend the definition of norm to all non-zerofractional ideals by defining
N(I) = N(P1)e1 · · ·N(Pr )er
for all non-zero fractional ideals I as in (1). With this extended definitionthe norm is a group homomorphism from the non-zero fractional ideals tothe positive rational numbers. Moreover, it “does the right thing” inregards to indices and quotient rings. See exercise #2 from homework #4.
Math 681, Monday, February 1 February 1, 2021
![Page 135: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/135.jpg)
Given a prime number p ∈ Z,
we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 136: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/136.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 137: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/137.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 138: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/138.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p.
We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 139: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/139.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p.
An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 140: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/140.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p,
thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 141: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/141.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi .
Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 142: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/142.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q].
Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 143: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/143.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 144: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/144.jpg)
Given a prime number p ∈ Z, we apply the Fundamental Theorem to theprincipal ideal generated by p in OK :
pOK = Pe11 · · ·P
err . (5)
Note that the non-zero prime ideals P1, . . . ,Pr here are precisely thoseprime ideals of OK that contain the prime number p. We say these primeideals lie above p. An earlier exercise showed that OK/Pi was a finite fieldof characteristic p, thus is the finite field with pfi elements for somepositive integer fi . Another exercise applied to the principal ideal pOK
showed that N(pOK ) = |NK/Q(p)| = pn, where n = [K : Q]. Therefore bythe Theorem above and equation (5),
[K : Q] = n = e1f1 + · · ·+ er fr .
Math 681, Monday, February 1 February 1, 2021
![Page 145: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/145.jpg)
Definition
The exponents ei in (5) are called the ramification indices of the primeideals Pi . If ei > 1 for any i we say the prime number p ramifies in thenumber field K . The positive integers fi are called the residue classdegrees or inertial degrees of the prime ideals Pi .
Obviously an important task is to determine the ramification indices andresidue class degrees.
We’ll work hard to show that the prime numbers p that ramify areprecisely the primes dividing the discriminant. Thus the ramification indexis equal to 1 with finitely many exceptions.
Math 681, Monday, February 1 February 1, 2021
![Page 146: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/146.jpg)
Definition
The exponents ei in (5) are called the ramification indices of the primeideals Pi .
If ei > 1 for any i we say the prime number p ramifies in thenumber field K . The positive integers fi are called the residue classdegrees or inertial degrees of the prime ideals Pi .
Obviously an important task is to determine the ramification indices andresidue class degrees.
We’ll work hard to show that the prime numbers p that ramify areprecisely the primes dividing the discriminant. Thus the ramification indexis equal to 1 with finitely many exceptions.
Math 681, Monday, February 1 February 1, 2021
![Page 147: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/147.jpg)
Definition
The exponents ei in (5) are called the ramification indices of the primeideals Pi . If ei > 1 for any i we say the prime number p ramifies in thenumber field K .
The positive integers fi are called the residue classdegrees or inertial degrees of the prime ideals Pi .
Obviously an important task is to determine the ramification indices andresidue class degrees.
We’ll work hard to show that the prime numbers p that ramify areprecisely the primes dividing the discriminant. Thus the ramification indexis equal to 1 with finitely many exceptions.
Math 681, Monday, February 1 February 1, 2021
![Page 148: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/148.jpg)
Definition
The exponents ei in (5) are called the ramification indices of the primeideals Pi . If ei > 1 for any i we say the prime number p ramifies in thenumber field K . The positive integers fi are called the residue classdegrees or inertial degrees of the prime ideals Pi .
Obviously an important task is to determine the ramification indices andresidue class degrees.
We’ll work hard to show that the prime numbers p that ramify areprecisely the primes dividing the discriminant. Thus the ramification indexis equal to 1 with finitely many exceptions.
Math 681, Monday, February 1 February 1, 2021
![Page 149: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/149.jpg)
Definition
The exponents ei in (5) are called the ramification indices of the primeideals Pi . If ei > 1 for any i we say the prime number p ramifies in thenumber field K . The positive integers fi are called the residue classdegrees or inertial degrees of the prime ideals Pi .
Obviously an important task is to determine the ramification indices andresidue class degrees.
We’ll work hard to show that the prime numbers p that ramify areprecisely the primes dividing the discriminant. Thus the ramification indexis equal to 1 with finitely many exceptions.
Math 681, Monday, February 1 February 1, 2021
![Page 150: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/150.jpg)
Definition
The exponents ei in (5) are called the ramification indices of the primeideals Pi . If ei > 1 for any i we say the prime number p ramifies in thenumber field K . The positive integers fi are called the residue classdegrees or inertial degrees of the prime ideals Pi .
Obviously an important task is to determine the ramification indices andresidue class degrees.
We’ll work hard to show that the prime numbers p that ramify areprecisely the primes dividing the discriminant.
Thus the ramification indexis equal to 1 with finitely many exceptions.
Math 681, Monday, February 1 February 1, 2021
![Page 151: February 1, 2021 · 2021. 2. 2. · For a non-zero fractional ideal I as in (1) above, the order of I at the maximal ideal P i is e i for i = 1;:::;r. For all other maximal ideals](https://reader036.vdocuments.us/reader036/viewer/2022071417/61149bb753f98461bd27b391/html5/thumbnails/151.jpg)
Definition
The exponents ei in (5) are called the ramification indices of the primeideals Pi . If ei > 1 for any i we say the prime number p ramifies in thenumber field K . The positive integers fi are called the residue classdegrees or inertial degrees of the prime ideals Pi .
Obviously an important task is to determine the ramification indices andresidue class degrees.
We’ll work hard to show that the prime numbers p that ramify areprecisely the primes dividing the discriminant. Thus the ramification indexis equal to 1 with finitely many exceptions.
Math 681, Monday, February 1 February 1, 2021