38 1. Elementary Prime Number Theory, I
16. (Continuation; Kronecker [Kro88], Dirichlet, Bauer [Bau06]) Define
Φm(X, Y ) as the homogenization of Φm(T), so that
Φm(X, Y ) =
=1 if 1 ≤ j m
(X − ζY ).
(a) Suppose m 2. Show that Φm(X + Y, X − Y ) = Gm(X, Y
some polynomial Gm (say) with integer coeﬃcients. Show also that
is the coeﬃcient of
in Φm(X + Y, X − Y ).
(b) Let F be a field of characteristic not dividing m. Suppose s is a
nonsquare integer, and let
s denote a fixed square root of s from
the algebraic closure of F . Show that the roots of Gm(T, s) ∈ Z[T]
in the algebraic closure of F are precisely the elements
ζ + 1
ζ − 1
where ζ runs through the primitive mth roots of unity.
(c) Suppose s is as in (b), and let p be a prime for which p(s2ms. Show
that p is a prime divisor of Gm(T, s) if and only if p ≡
(d) Show that if p ≡ −1 (mod 4) is a prime divisor of Gm(T, −1) which
does not divide m, then p ≡ −1 (mod m). Use Exercise 14 to
show that Gm(T, −1) has infinitely many such prime divisors, and
deduce that there are infinitely many primes p ≡ −1 (mod m).
17. (M. Hirschhorn [Hir02]) Let p1 p2 p3 · · · denote the sequence of
(a) Let N ∈ N. Prove that the number of odd positive integers ≤ N
which can be written in the form p11
· · · pkk
does not exceed
+ 1 (log
Hint: Show that (log
whenever u ≥ 1. Now
invoke the inequality m! ≥
valid for every integer m ≥ 0.
(b) Supposing that p1,...,pk exist (i.e., that there are at least k odd
primes), prove that pk+1 exists and satisfies pk+1 ≤ 4(2k!)pk + 1.
18. Suppose that A is a commutative monoid (written multiplicatively) and
that P is a system of generators for A, so that each element of A can be
written in the form
, where each ep ≥ 0 and only finitely many
of the ep are nonzero. (We do not require that this representation be
unique.) Suppose also that there is a function · : A → N with the
following two properties:
(a) · respects multiplication, i.e., ab = a b for all a, b ∈ A.