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 ) = ζm=1 ζj=1 if 1 ≤ j m (X − ζY ). (a) Suppose m 2. Show that Φm(X + Y, X − Y ) = Gm(X, Y 2 ) for some polynomial Gm (say) with integer coeﬃcients. Show also that d|m dµ(m/d) is the coeﬃcient of Xϕ(m) 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 √ s 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 2ms. Show that p is a prime divisor of Gm(T, s) if and only if p ≡ ( s p (mod m). (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 odd primes. (a) Let N ∈ N. Prove that the number of odd positive integers ≤ N which can be written in the form pe1 1 · · · pek k does not exceed k i=1 log N log pi + 1 (log (pkN))k √ 2k! pkN. Hint: Show that (log u)ku−1/2 ≤ (2k/e)k whenever u ≥ 1. Now invoke the inequality m! ≥ (m/e)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 p∈P pep, 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.

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