Exercises 39

(b) For some real number x0 and constants c1,c2 0, we have

(1.18) c1x ≤ #{a ∈ A : a ≤ x} ≤ c2x for all x x0.

Prove that P is infinite, and that in fact

∑

p∈P

1

p

diverges.

19. (Continuation)

(a) For each nonzero Gaussian integer α put α =

|α|2.

Show that

∑

π

π −1 diverges, where the sum is over all Gaussian primes π.

Deduce that

∑

p≡1 (mod 4)

p−1

diverges, where the sum is over ra-

tional primes p ≡ 1 mod 4.

(b) For each nonzero polynomial F (T) ∈ Fq[T], put F :=

qdeg F

.

Show that

∑

P

−1

diverges, where P ranges over the irreducible

elements of Fq[T].

20. This exercise outlines a proof of Theorem 1.21 via algebraic number

theory. Let m be a positive integer, and let ζ be a primitive mth root

of unity. Put K = Q(ζm), and identify Gal(K/Q) with

(Z/mZ)×.

Let

H be a subgroup of

(Z/mZ)×,

and let L ⊂ K be the fixed field of H.

(a) Say that two sets of rational primes P1 and P2 eventually coincide

if their symmetric difference is finite; in this case we write P1

.

=

P2. Prove that P1

.

= P2, where P1 is the set of primes for which

p mod m ∈ H and P2 is the set of primes which split completely in

L. Hint: If p is a prime not dividing m, analyze how the Frobenius

element of p in Gal(K/Q) behaves upon restriction to L.

(b) Let θ be an algebraic integer for which L = Q(θ). Let F be the min-

imal polynomial of θ. Prove that P2, and hence also P1, eventually

coincides with the set of prime divisors of F . Hint: L/Q is Galois,

so an unramified rational prime splits completely in L exactly when

it has a degree 1 prime factor; now apply the Kummer-Dedekind

theorem.

21. (P´ olya [P´ ol21]; see also [MS00]) Suppose that a and b are nonzero in-

tegers and a = ±1. Let P be the set of primes for which the exponential

congruence

ak

≡ b (mod p) has a positive integer solution k. In other

words, P is the set of primes which divide some term of the sequence

a − b,

a2

− b,

a3

− b,

a4

− b, . . . .

This exercise outlines a proof that P is always an infinite set.

We may suppose that b is not a power of a, as otherwise P contains

every prime. We assume for the sake of contradiction that P is finite.

(a) For each p ∈ P and each k ≥ 1, define integers vp,k ≥ 0 by writing

ak

− b = ±

p∈P

pvp,k

.