26 1. Elementary Prime Number Theory, I
Theorem 1.21. Let m be a positive integer and let H be a subgroup of
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
≡ 1 (mod m), where a ≡ 1 (mod m).
Applying Theorem 1.21 to the 2-element subgroup of
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
≡ 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
≡ 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. 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),