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How Are Irreducible and Primitive PolynomialsDistributed over Finite Fields?
Gary L. Mullen
Penn State University
July 21, 2010
Gary L. Mullen (PSU) How Are Irreducible and Primitive Polynomials Distributed over Finite Fields?July 21, 2010 1 / 28
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Let q = pm be a prime power
Let Fq = Fpm denote the finite field of order q
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Primitive Polynomials
f ∈ Fq[x] of deg. n is primitive if every root of f is a prim. ele. in Fqn
Recall that a prim. ele. in Fqn generates the mul. group F ∗qn
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Theorem
Cohen (Disc. Math., 90) Let n ≥ 2 and let a ∈ Fq (with a 6= 0 if n = 2 orif n = 3 and q = 4). Then there exists prim. deg. n over Fq with trace a.
Conjecture
Hansen/M (Math. Comp, 92)Conj A: For n ≥ 2, 0 ≤ j < n and given a ∈ Fq, there is prim.xn + · · ·+ axj + · · · except when(P1) q arb., j = 0, a 6= (−1)nα,α prim. in Fq
(P2) q arb., n = 2, j = 1, a = 0(P3) q = 4, n = 3, j = 2, a = 0(P4) q = 4, n = 3, j = 1, a = 0(P5) q = 2, n = 4, j = 2, a = 1
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Theorem
Cohen (Disc. Math., 90) Let n ≥ 2 and let a ∈ Fq (with a 6= 0 if n = 2 orif n = 3 and q = 4). Then there exists prim. deg. n over Fq with trace a.
Conjecture
Hansen/M (Math. Comp, 92)Conj A: For n ≥ 2, 0 ≤ j < n and given a ∈ Fq, there is prim.xn + · · ·+ axj + · · · except when(P1) q arb., j = 0, a 6= (−1)nα,α prim. in Fq
(P2) q arb., n = 2, j = 1, a = 0(P3) q = 4, n = 3, j = 2, a = 0(P4) q = 4, n = 3, j = 1, a = 0(P5) q = 2, n = 4, j = 2, a = 1
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Many papers by various people culminating in
Theorem
Cohen (FFA 06) Conj. A is true for deg. n ≥ 9.
Theorem
Cohen/Presern (Lond. Math. Soc. Lect. Note Ser. 07) Conj. A is true!!
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Many papers by various people culminating in
Theorem
Cohen (FFA 06) Conj. A is true for deg. n ≥ 9.
Theorem
Cohen/Presern (Lond. Math. Soc. Lect. Note Ser. 07) Conj. A is true!!
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Other Related Results
Theorem
Han (Math. Comp., 96) For q odd and n ≥ 7, ∃ prim. deg. n with coeffs.of xn−1 and xn−2 given in advance.
Theorem
Han (FFA 97) For q even and n ≥ 7, ∃ prim. deg. n with coeff. xn−1 andxn−2 given in advance.
Theorem
Cohen/Mills (FFA 03) For q odd and n = 5, 6 ∃ prim. deg. n with coeff.xn−1 and xn−2 given in advance.
Theorem
Shuqin/Han (FFA 04) For n ≥ 8 ∃ prim. deg. n with highest three coeff.given in advance.
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Other Related Results
Theorem
Han (Math. Comp., 96) For q odd and n ≥ 7, ∃ prim. deg. n with coeffs.of xn−1 and xn−2 given in advance.
Theorem
Han (FFA 97) For q even and n ≥ 7, ∃ prim. deg. n with coeff. xn−1 andxn−2 given in advance.
Theorem
Cohen/Mills (FFA 03) For q odd and n = 5, 6 ∃ prim. deg. n with coeff.xn−1 and xn−2 given in advance.
Theorem
Shuqin/Han (FFA 04) For n ≥ 8 ∃ prim. deg. n with highest three coeff.given in advance.
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Other Related Results
Theorem
Han (Math. Comp., 96) For q odd and n ≥ 7, ∃ prim. deg. n with coeffs.of xn−1 and xn−2 given in advance.
Theorem
Han (FFA 97) For q even and n ≥ 7, ∃ prim. deg. n with coeff. xn−1 andxn−2 given in advance.
Theorem
Cohen/Mills (FFA 03) For q odd and n = 5, 6 ∃ prim. deg. n with coeff.xn−1 and xn−2 given in advance.
Theorem
Shuqin/Han (FFA 04) For n ≥ 8 ∃ prim. deg. n with highest three coeff.given in advance.
Gary L. Mullen (PSU) How Are Irreducible and Primitive Polynomials Distributed over Finite Fields?July 21, 2010 6 / 28
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Other Related Results
Theorem
Han (Math. Comp., 96) For q odd and n ≥ 7, ∃ prim. deg. n with coeffs.of xn−1 and xn−2 given in advance.
Theorem
Han (FFA 97) For q even and n ≥ 7, ∃ prim. deg. n with coeff. xn−1 andxn−2 given in advance.
Theorem
Cohen/Mills (FFA 03) For q odd and n = 5, 6 ∃ prim. deg. n with coeff.xn−1 and xn−2 given in advance.
Theorem
Shuqin/Han (FFA 04) For n ≥ 8 ∃ prim. deg. n with highest three coeff.given in advance.
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Problem
Find formulas, or good estimates, for the # of prim. deg. n over Fq withgiven trace, (or even more coeff). specified in advance.
Theorem
Chang/Chou/Shiue (FFA 05) Enum. results.
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Problem
Find formulas, or good estimates, for the # of prim. deg. n over Fq withgiven trace, (or even more coeff). specified in advance.
Theorem
Chang/Chou/Shiue (FFA 05) Enum. results.
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Primitive Normal Polynomials
For α ∈ Fqn , if A = {α, αq, . . . , αqn−1} is a basis over Fq, A is normalbasis.
If 〈α〉 = F ∗qn , A is prim. nor. basis.
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Theorem
Carlitz (Trans. 52) Prim. nor. basis over Fp for suff. large p.
Theorem
Davenport (J. Lond. M. S. 68) Prim. nor. basis over Fp for any prime p.
Theorem
Lenstra-Schoof (Math. Comp., 87) Fqn has prim. nor. basis over Fq.
Theorem
Cohen/Huczynska (J. London. M. Soc. 03) Prim. nor. basis without acomputer!
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Theorem
Carlitz (Trans. 52) Prim. nor. basis over Fp for suff. large p.
Theorem
Davenport (J. Lond. M. S. 68) Prim. nor. basis over Fp for any prime p.
Theorem
Lenstra-Schoof (Math. Comp., 87) Fqn has prim. nor. basis over Fq.
Theorem
Cohen/Huczynska (J. London. M. Soc. 03) Prim. nor. basis without acomputer!
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Theorem
Carlitz (Trans. 52) Prim. nor. basis over Fp for suff. large p.
Theorem
Davenport (J. Lond. M. S. 68) Prim. nor. basis over Fp for any prime p.
Theorem
Lenstra-Schoof (Math. Comp., 87) Fqn has prim. nor. basis over Fq.
Theorem
Cohen/Huczynska (J. London. M. Soc. 03) Prim. nor. basis without acomputer!
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Theorem
Carlitz (Trans. 52) Prim. nor. basis over Fp for suff. large p.
Theorem
Davenport (J. Lond. M. S. 68) Prim. nor. basis over Fp for any prime p.
Theorem
Lenstra-Schoof (Math. Comp., 87) Fqn has prim. nor. basis over Fq.
Theorem
Cohen/Huczynska (J. London. M. Soc. 03) Prim. nor. basis without acomputer!
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φ(qn − 1) is # of prim. elem. in Fqn
Φq(xn − 1) is # of nor. basis elem. in Fqn .
Problem
Find a formula for the number PNq(n) of prim. nor. elem. in Fqn .
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Conjecture
Morgan/Mul. (Math. Comp., 94) For n ≥ 2 and a ∈ F ∗q , there is a prim.nor. poly. deg. n over Fq with trace a.
True for:
q = 2 any n
n = 2 any q
(q − 1)|n
n ≤ 6, q ≤ 97
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Theorem
Cohen/Hac. (AAECC, 99) Morgan/M Conj. true
Conjecture
Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n ≥ 2, ∃ prim.nor. poly. deg. n over Fp with at most 5 nonzero coeffs.
Theorem
Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n ≥ 5 with given normand trace.
Theorem
Cohen/Huczynska (Acta Arith, 03) Prim. nor. quartics with given normand trace
Theorem
Huczynska/Cohen (Trans. A.M.S. 03) Prim. nor. cubics with givennorm and trace
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Theorem
Cohen/Hac. (AAECC, 99) Morgan/M Conj. true
Conjecture
Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n ≥ 2, ∃ prim.nor. poly. deg. n over Fp with at most 5 nonzero coeffs.
Theorem
Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n ≥ 5 with given normand trace.
Theorem
Cohen/Huczynska (Acta Arith, 03) Prim. nor. quartics with given normand trace
Theorem
Huczynska/Cohen (Trans. A.M.S. 03) Prim. nor. cubics with givennorm and trace
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Theorem
Cohen/Hac. (AAECC, 99) Morgan/M Conj. true
Conjecture
Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n ≥ 2, ∃ prim.nor. poly. deg. n over Fp with at most 5 nonzero coeffs.
Theorem
Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n ≥ 5 with given normand trace.
Theorem
Cohen/Huczynska (Acta Arith, 03) Prim. nor. quartics with given normand trace
Theorem
Huczynska/Cohen (Trans. A.M.S. 03) Prim. nor. cubics with givennorm and trace
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Theorem
Cohen/Hac. (AAECC, 99) Morgan/M Conj. true
Conjecture
Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n ≥ 2, ∃ prim.nor. poly. deg. n over Fp with at most 5 nonzero coeffs.
Theorem
Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n ≥ 5 with given normand trace.
Theorem
Cohen/Huczynska (Acta Arith, 03) Prim. nor. quartics with given normand trace
Theorem
Huczynska/Cohen (Trans. A.M.S. 03) Prim. nor. cubics with givennorm and trace
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Theorem
Cohen/Hac. (AAECC, 99) Morgan/M Conj. true
Conjecture
Morgan/Mul. Conj: (Math. Comp., 94) For p prime and n ≥ 2, ∃ prim.nor. poly. deg. n over Fp with at most 5 nonzero coeffs.
Theorem
Cohen (Pub. Math. Deb. 00) Prim. nor. of deg. n ≥ 5 with given normand trace.
Theorem
Cohen/Huczynska (Acta Arith, 03) Prim. nor. quartics with given normand trace
Theorem
Huczynska/Cohen (Trans. A.M.S. 03) Prim. nor. cubics with givennorm and trace
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Theorem
Fan/Wang (FFA 09) If n ≥ 15, ∃ prim. nor. with any coeff. specified inadvance
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Completely Normal Bases
Fq ⊆ Fqd ⊆ Fqn
∃α ∈ Fqn nor. basis over Fq and Fqd ?
∃α ∈ Fqn nor. basis over Fqd for all d|n ?
Theorem
Blessenohl/Johnsen (J. Alg., 86) Fqn has a com. nor. basis.
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Conjecture
Morgan/M (Util. Math. 96) For each n ≥ 2 there is a com. nor. prim.poly. deg. n over Fq.
True for: n = 4
qn ≤ 231, q ≤ 97
Theorem
Shparlinski/Mul. (Finite Fields Appl., CUP, 96) For q ≥ Cnlog n, ∃com. nor. prim. basis of Fqn over Fq.
Problem
Find formula for the number CNq(n) of com. nor. bases of Fqn over Fq.
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Conjecture
Morgan/M (Util. Math. 96) For each n ≥ 2 there is a com. nor. prim.poly. deg. n over Fq.
True for: n = 4
qn ≤ 231, q ≤ 97
Theorem
Shparlinski/Mul. (Finite Fields Appl., CUP, 96) For q ≥ Cnlog n, ∃com. nor. prim. basis of Fqn over Fq.
Problem
Find formula for the number CNq(n) of com. nor. bases of Fqn over Fq.
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Irreducibles
Conjecture
Hansen/M (Math. Comp. 92)Conj B: For n ≥ 2, 0 ≤ j < n and given a ∈ Fq there is irr.xn + · · ·+ axj + · · · except(I1) q arb. j = a = 0(I2) q = 2m, n = 2, j = 1, a = 0
Theorem
Wan (Math. Comp., 97) If either q > 19 or n ≥ 36, Conj. B is true.
Theorem
Ham/Mul. (Math. Comp. 97) Conj. B is true.
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Irreducibles
Conjecture
Hansen/M (Math. Comp. 92)Conj B: For n ≥ 2, 0 ≤ j < n and given a ∈ Fq there is irr.xn + · · ·+ axj + · · · except(I1) q arb. j = a = 0(I2) q = 2m, n = 2, j = 1, a = 0
Theorem
Wan (Math. Comp., 97) If either q > 19 or n ≥ 36, Conj. B is true.
Theorem
Ham/Mul. (Math. Comp. 97) Conj. B is true.
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Irreducibles
Conjecture
Hansen/M (Math. Comp. 92)Conj B: For n ≥ 2, 0 ≤ j < n and given a ∈ Fq there is irr.xn + · · ·+ axj + · · · except(I1) q arb. j = a = 0(I2) q = 2m, n = 2, j = 1, a = 0
Theorem
Wan (Math. Comp., 97) If either q > 19 or n ≥ 36, Conj. B is true.
Theorem
Ham/Mul. (Math. Comp. 97) Conj. B is true.
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Theorem
Hsu (J. Numb. Thy., 96) (i) If f has even deg. n and q ≥ n/2 + 1, ∃ irr.P of deg. n with deg. (P − f) at most n/2.(ii) If f has odd deg. n and q ≥ ((n+ 3)/2)2, ∃ irr. P of deg. n with deg.(P − f) at most (n− 1)/2.
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Exact Formulas
Nq(n) = # monic irr. deg. n
=1n
∑d|n
µ(d)qn/d
# monic irr. deg. n trace 1
12n
∑d|n,dodd
µ(d)2n/d
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Fix 1 ≤ j ≤ n, β ∈ F2n
Tj(β) =∑
0≤i1<i2<···<ij≤n
β2i1β2i2 · · ·β2ij
Tj : F2n → F2
T1(β) = β + β2 + β22+ · · ·+ β2n−1
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F (n, t1, . . . , tr) = #β ∈ F2n with Tj(β) = tj , j = 1, . . . , r
P (n, t1, . . . , tr) = # irr. deg n with coeff. xn−j = tj , j = 1, . . . , r
P (n, 0, 0, 1) = #xn + 0xn−1 + 0xn−2 + 1xn−3 + . . .
nP (n, 0, 0, 1) =∑
d|n,dodd
µ(d)F (n/d, 0, 0, 1)
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Theorem
Cattell/Miers/Ruskey/Serra/Sawada (JCMCC 03)F (n, t1, t2) = 2n−2 +G(n, t1, t2),
G(n, t1, t2) =
m(mod4) 00 01 10 11
0 −2m−1 2m−1 0 01 0 0 −2m−1 2m−1
2 2m−1 −2m−1 0 03 0 0 2m−1 −2m−1
Theorem
Kuz’min (Sov. Math. Dokl. 91)
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Theorem
Yucas/M (Disc. Math. 04) For n even,F (n, t1, t2, t3) = 2n−3 +G(n, t1, t2, t3),
m(mod12) 000 001 010 011
0 −2m − 2m−2 2m−1 + 2m−2 2m−2 2m−2
1or5 2m−2 −2m−2 2m−2 −2m−2
2or10 0 2m−1 0 −2m−1
3 2m−2 −2m−2 2m−2 −2m−2
4or8 −2m−1 0 −2m−1 2m
6 2m−1 + 2m−2 −2m−2 −2m−1 − 2m−2 2m−2
7or11 2m−2 −2m−2 2m−2 −2m−2
9 2m−2 −2m−2 2m−2 −2m−2
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Theorem
m(mod12) 100 101 110 111
0 0 0 0 01or5 −2m−2 −2m−2 2m−1 + 2m−2 −2m−2
2or10 2m−1 −2m−1 −2m−1 2m−1
3 2m−1 0 2m−1 −2m
4or8 0 0 0 06 2m−1 −2m−1 −2m−1 2m−1
7or11 −2m−2 2m−1 + 2m−2 −2m−2 −2m−2
9 −2m 2m−1 0 2m−1
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Conjecture
Yucas/Mul If n = 2m, F (n, t1, . . . , tr) =
2n−r + am−s+12m−s+1 + · · ·+ am2m
1 ≤ s ≤ m, ai = −1, 0, 1
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Theorem
Fitzgerald/Yucas (FFA 03) Formula for F (n, t1, t2, t3) for odd n.non-deg., alt., sym., bil., quad. forms
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Problem
Extend to more than three coeff. over F2.
Problem
Extend two coeff. case to Fq for q ≥ 2.
Problem
Extend to j coeff. over Fq.
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Problem
Extend to more than three coeff. over F2.
Problem
Extend two coeff. case to Fq for q ≥ 2.
Problem
Extend to j coeff. over Fq.
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Problem
Extend to more than three coeff. over F2.
Problem
Extend two coeff. case to Fq for q ≥ 2.
Problem
Extend to j coeff. over Fq.
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Several Variables
Theorem
Corteel/Savage/Wilf/Zeilberger, (JCT,A 98) The # of pairs of polys.f(x) and g(x) of deg. m over F2 with (f, g) = 1 is the same as the # ofpairs of polys. of deg. m with (f, g) 6= 1.
Theorem
Benjamin/Bennett, (Math. Mag. 07), Euclid algor. biject.
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Several Variables
Theorem
Corteel/Savage/Wilf/Zeilberger, (JCT,A 98) The # of pairs of polys.f(x) and g(x) of deg. m over F2 with (f, g) = 1 is the same as the # ofpairs of polys. of deg. m with (f, g) 6= 1.
Theorem
Benjamin/Bennett, (Math. Mag. 07), Euclid algor. biject.
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Let f ∈ Fq[x1, . . . , xk] with k ≥ 2
Two notions: total deg. and vector deg. of f
Problem
Count # of irr. polys. of a given deg. in Fq[x1, . . . , xk]
Problem
Count # of pairs of relatively prime polys. of a given deg. inFq[x1, . . . , xk]
Each problem has a total deg. version and a vector deg. version
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Let f ∈ Fq[x1, . . . , xk] with k ≥ 2
Two notions: total deg. and vector deg. of f
Problem
Count # of irr. polys. of a given deg. in Fq[x1, . . . , xk]
Problem
Count # of pairs of relatively prime polys. of a given deg. inFq[x1, . . . , xk]
Each problem has a total deg. version and a vector deg. version
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Hou/Mul. (FFA 09) Results for several variables
Bodin (FFA 10) Generating series for # irr. of given deg. and forindecomp. polys.
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Hou/Mul. (FFA 09) Results for several variables
Bodin (FFA 10) Generating series for # irr. of given deg. and forindecomp. polys.
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