the p-adic number system - university of minnesotaemanlove/p-adic presentation.pdf · background...
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![Page 1: The p-adic Number System - University of Minnesotaemanlove/p-adic presentation.pdf · Background The p-adic numbers were introduced by Kurt Hensel in 1897. Since then, they have found](https://reader033.vdocuments.us/reader033/viewer/2022052709/5a7911457f8b9a07628be4c8/html5/thumbnails/1.jpg)
The p-adic Number System
Erin Manlove
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Background
The p-adic numberswere introduced byKurt Hensel in 1897.
Since then, they have found their way into manyother areas of mathematics and even physics.
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Observation
Fix a prime p. Any rational number st with s, t ∈ Z
can be multiplied by some power of p to get a newrational number s∗
t∗ where GCD(s∗, p) = 1 andGCD(t∗, p) = 1.
For example, let p = 3.
900 · 3−2 = 100
−1823 · 3
−2 = − 223
521 · 3 = 5
7
14 · 3
0 = 14
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Definition: The p-adic Valuation
Define | · |p : Q→ R+ ∪ {0} by
|0|p = 0
and for r , s 6= 0, ∣∣∣∣rs∣∣∣∣p
= pn
where pn · rs = r∗
s∗ with GCF(r ∗, p) = 1 andGCF(s∗, p) = 1.
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3-adic Examples
Again, let p = 3.
900 · 3−2 = 100
|900|3 = 3−2
−1823 · 3
−2 = − 223
| − 1823|3 = 3−2
521 · 3 = 5
7
| 521|3 = 3
14 · 3
0 = 14
|14 |3 = 1
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Properties of the p-adic Valuation
1. |a|p = 1 if and only if a = rs with
GCF(p, r) = 1 =GCF(p, s).
2. |a|p = pn if and only if |pna|p = 1.
3. For every integer n, |pna|p = p−n|a|p.Let a 6= 0 and |a|p = pm.Then from (2), |pma|p = 1.Rewrite this as |pm−npna|p = 1.Again from (2), |pna|p = pm−n= p−n|a|p.
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Back up: can we say that it’s a valuation?
A valuation of Q is a map | · | : Q→ R+ ∪ {0}which satisfies 3 properties:
(i) |0| = 0; |a| > 0 if a 6= 0
We said that |0|p = 0 and
∣∣∣∣ rs ∣∣∣∣p
= 3n > 0.
(ii) |ab| = |a| · |b|
To eliminate all factors of p from ab we couldmultiply by a power of p to remove them from a,then multiply by a power of p to remove them fromb.
(iii) |a + b| ≤ |a|+ |b|
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Third Property of Valuations
In fact, we’ll prove something BETTER:
|a + b|p ≤ max{|a|p, |b|p}
Proof: If a = 0 or b = 0, then this is clear.Let a, b 6= 0 have p-adic values |a|p = pn and|b|p = pm, where n ≤ m.The denominators of both pna and pmb have nofactors of p, and so neither does the denominator ofpm(a + b).That means |pm(a + b)|p ≤ 1.Then from property (3), p−m|a + b|p ≤ 1.|a + b|p ≤ pm = max{|a|p, |b|p}.
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Series for Rational Numbers
Claim: Any rational number can be written in theform
∑∞k=n akp
k , where ak ∈ {0, . . . , p − 1}.
Examples:
1
3= 1 · 3−1 + 0 · 30 + 0 · 31 + 0 · 32 + . . .
1
3= 10, 000 . . .3
25 = 1 · 30 + 2 · 31 + 2 · 32 + 0 · 33 + 0 · 34 + . . .
25 = 1, 22000 . . .3
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More Interesting Example
2
5= 1 + 3 · −1
5
−1
5= 1 + 3 · −2
5−2
5= 2 + 3 · −4
5
−4
5= 1 + 3 · −3
5
−3
5= 0 + 3 · −1
5−1
5= 1 + 3 · −2
5. . .
2
5= 30(1)+31(1)+32(2)+33(1)+34(0)+35(1)+. . .
2
5= 1, 12101210 . . .3
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Let’s Build Spaces!
The p-adic Numbers Qp are sequences
a =∞∑k=n
akpk ,
where ak ∈ {0, . . . , p − 1}.
The p-adic Integers Zp are elements a ∈ Qp suchthat |a|p ≤ 1.
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Visualization
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Visualization
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Visualization
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Sierpinski Triangle
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References
I Mahler, K. P-adic Numbers and TheirFunctions, 2nd ed. Cambridge, England:Cambridge University Press, 1981.
I Albert A. Cuoco. Visualizing the p-adic integers.Amer. Math. Monthly, 98:355364, 1991.