Prime numbers and prime polynomials, Exams of Number Theory

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Prime numbers and prime

polynomials

Paul Pollack Dartmouth College

May 1, 2008

Analogies everywhere!

• Analogies in elementary number theory (con- tinued fractions, quadratic reciprocity, Fer- mat’s last theorem)
• Analogies in algebraic number theory (the theory of global function fields vs. the the- ory of algebraic number fields)
• Analogies in analytic number theory, espe- cially prime number theory

Prime number theorem (Hadamard, de la Vall´ee Poussin). If π(x) denotes the number of primes p ≤ x, then

π(x) ∼ x log x

as x → ∞.

Prime number theorem for polynomials. Let π(q; d) denote the number of monic, degree d

irreducibles over the finite field Fq. Then as

qd^ → ∞, we have

π(q; d) ∼ qd d

Notice that if X = qd, then qd/d = X/ logq X.

Gauss’s take on the prime number theorem: Empirical observations suggest that the primes near x have a density of about 1/ log x. So we should have

π(x) ≈

log 2

log 3

log x

Theorem (von Koch). If the Riemann Hypoth- esis is true, then

π(x) =

∑ 2 ≤n≤x

log n

• O(x^1 /^2 log x).

Define πp(X) as the number of n ≤ x which

encode irreducible polynomials over Fp.

We might hope that

πp(X) ≈

∑ ||f ||≤x

deg f

Theorem. If X ≥ p, then

πp(X) =

∑ ||f ||≤x

deg f

+ O

( dpd/2+

) ,

where pd^ ≤ X < pd+1.

Notice dpd/2+1^ p X^1 /^2 log X, so this is a von Koch analogue.

Proof idea: To each global function field K

(finite extension of Fq(T )) one associates a

zeta function,

ζK(s) =

∑ a≥ 0

Nm(a)s^

A deep theorem of Weil asserts that these zeta functions all satisfy the analogue of the Rie- mann Hypothesis.

Define L-functions which are sensitive to the behavior of the initial coefficients of a polyno-

mial in Fq[T ]. The analytic properties of this

L-function can then be linked to the analytic properties of ζK(s) for an appropriate K and the Riemann Hypothesis brought into play.

Theorem (Hall). Suppose q > 3. Then there are infinitely many monic irreducibles P (T ) over

Fq for which P (T ) + 1 is also irreducible.

Theorem (P.). Suppose q > 3. Then there are

infinitely many monic irreducibles P (T ) over Fq

for which P (T ) + 1 is also irreducible.

Theorem (Capelli’s Theorem). Let F be any field. The binomial T m^ − a is reducible over F if and only if either of the following holds:

• there is a prime l dividing m for which a is an lth power in F ,
• 4 divides m and a = − 4 b^4 for some b in F.

Observe: We have

x^4 + 4y^4 = (x^2 + 2y^2 )^2 − (2xy)^2.

A finite field analogue of Hypothesis H. Suppose f 1 ,... , fr are irreducible polynomials

in Fq[T ] and that there is no irreducible P in

Fq[T ] for which

P (T ) always divides f 1 (h(T )) · · · fr(h(T )).

Then f 1 (h(T )),... , fr(h(T )) are simultaneously irreducible for infinitely many monic polynomi-

als h(T ) ∈ Fq[T ].

Example: “Twin prime” pairs: take f 1 (T ) := T and f 2 (T ) := T + 1.

Observation: The local condition is always sat- isfied if

q >

∑^ r i=

deg fi.

Theorem (P.). Suppose f 1 ,... , fr are irreducible

polynomials in Fq[T ]. Let D =

∑r i=1 deg^ fi. If q > max{ 3 , 22 r−^2 D^2 },

then there are infinitely many monic polynomi- als h(T ) for which all of f 1 (h(T )),... , fr(h(T )) are simultaneously irreducible.

Choose any prime l dividing q^2 − 1, and let let

χ be an lth power-residue character on Fq 2. If

there is no such β, then ∑ β∈Fq

χ(β + i) = q.

But Weil’s Riemann Hypothesis gives a bound for this incomplete character sum of √q – a contradiction.

Quantitative problems and results

Twin prime conjecture (quantitative version). The number of prime pairs p, p + 2 with p ≤ x is asymptotically

2 C 2 x log^2 x

as x → ∞,

where C 2 = ∏ p> 2 (1 − 1 /(p − 1)^2 ).

Can generalize to the full Hypothesis H situa- tion (Hardy-Littlewood/Bateman-Horn).

Theorem. Let n be a positive integer. Let f 1 (T ),... , fr(T ) be pairwise nonassociated ir-

reducible polynomials over Fq with the degree

of the product f 1 · · · fr bounded by B.

The number of univariate monic polynomials h of degree n for which all of f 1 (h(T )),... , fr(h(T ))

are irreducible over Fq is

qn/nr^ + O((nB)n!Bqn−^1 /^2 )

provided gcd(q, 2 n) = 1.

Example: The number of monic polynomials

h(T ) of degree 3 over Fq for which h(T )^2 + 1

is irreducible is asymptotically q^3 /3 as q → ∞ with q ≡ 3 (mod 4) and (q, 3) = 1.

Some ideas of the proof

The inspiration:

Conjecture (Chowla, 1966). Fix a positive in- teger n. Then for all large primes p, there is

always an irreducible polynomial in Fp[T ] of the

form T n^ + T + a with a ∈ Fp.

In fact, for fixed n the number of such a is asymptotic to p/n as p → ∞.

Proved by Ree and Cohen (independently) in