26 1. Elementary Prime Number Theory, I

Theorem 1.21. Let m be a positive integer and let H be a subgroup of

(Z/mZ)×.

There is a nonconstant polynomial F (T) ∈ Z[T] with the fol-

lowing property: Every prime divisor p of F , with finitely many exceptions,

satisfies p mod m ∈ H. Consequently, there are infinitely many primes p

for which p mod m ∈ H.

When H is the trivial subgroup we have just seen that F := Φm satisfies

the conclusion of Theorem 1.21.

Schur gave an elementary proof of Theorem 1.21 requiring only famil-

iarity with the theory of finite fields. A less elementary proof is outlined in

Exercise 20. When m is a prime number, Theorem 1.21 is contained in the

results of Chapter 2 (see, in particular, Theorem 2.15).

Suppose that a and m satisfy

a2

≡ 1 (mod m), where a ≡ 1 (mod m).

Applying Theorem 1.21 to the 2-element subgroup of

(Z/mZ)×

generated

by a mod m, we obtain a polynomial F (T) all of whose prime divisors (with

finitely many exceptions) satisfy either p ≡ 1 (mod m) or p ≡ a (mod m).

Schur showed (op. cit.) that if there is a single, suitably large prime p ≡

a (mod m), then the polynomial F he constructs cannot have all (or even

all but finitely many) of its prime divisors from the progression 1 mod m.

(See the first example below for an illustration of how this works.) So F

must have infinitely many prime divisors p ≡ a (mod m).

Since Dirichlet’s theorem is true, there is always a suitably large prime

p ≡ a (mod m) to be used in Schur’s argument, and so in principle, it

is possible to give a purely algebraic proof of Dirichlet’s theorem for any

progression a mod m satisfying

a2

≡ 1 (mod m). Moreover, this is best

possible in the following sense:

Theorem 1.22 (Murty [Mur88, MT06]). Suppose m is a positive in-

teger. If F is a nonconstant polynomial with the property that every prime

divisor p of F , with finitely many exceptions, satisfies p ≡ 1 (mod m) or

p ≡ a (mod m), then

a2

≡ 1 (mod m).

The proof of Theorem 1.22 rests on rather deep results in algebro-

analytic number theory. The principal tool required is the Chebotarev den-

sity theorem, which is a far-reaching generalization of Dirichlet’s theorem.

See [SL96] for a down-to-earth discussion of Chebotarev’s result.

Example. As an easy example of Schur’s method, consider the problem of

showing that there are infinitely many primes p ≡ 3 (mod 8). We start by

taking F (T) := T

2

+2. From the elementary theory of quadratic residues we

have that each odd prime divisor of F (T) satisfies p ≡ 1 or 3 (mod 8). Now

we observe that there is at least one prime in the residue class 3 (mod 8),