Exercises 37 10. (Ramanujan) Assuming ζ(2) = π2/6 and ζ(4) = π4/90, show that 1 n2 = 9 2π2 , where the indicates that the sum ranges over positive squarefree inte- gers n with an odd number of prime divisors. 11. (Cf. Porubsk´ y [Por01]) If R is a commutative ring, its Jacobson radical J(R) is the intersection of all of its maximal ideals. Show that J(R) = {x R : 1 xy is invertible for all y R}. Deduce that if R is an integral domain with finitely many units, then J(R) = {0}. Use this to prove that if R is a principal ideal domain with finitely many units, then either R is a field or R contains an infinite set of pairwise nonassociated primes. 12. By carefully examining the proof of Theorem 1.10, show that the the- orem remains correct when A = 1, provided that in condition (ii) we replace “prime” with “prime or equal to 1”. 13. Suppose that a1 a2 a3 · · · is an increasing sequence of natural numbers, and put A(x) := ai≤x 1. Prove that if (log x)−kA(x) for each fixed k, then infinitely many primes p divide some ai. Use this to give another proof of Lemma 1.17. 14. Prove the following theorem of Bauer [Bau06]: Theorem. If F (T) Z[T] is a nonconstant polynomial with at least one real root, then for every m 3, there exist infinitely many prime divisors p of F with p 1 (mod m). Proceed by showing that each of the following conditions on F is sufficient for the conclusion of the theorem to hold: (a) F has a positive leading coefficient and constant term −1. (b) F has a positive leading coefficient and negative constant term. (c) F has a positive leading coefficient and F (a) 0 for some a Z. (d) F has a positive leading coefficient and F (a) 0 for some a Q. (e) F has a positive leading coefficient and F (a) 0 for some a R. (f) F has a positive leading coefficient and F (a) = 0 for some a R. Hint for (f): Reduce to the case when F has no multiple roots. 15. Let F be a field of characteristic not dividing m. By carefully examining the proof of Lemma 1.19, show that the roots of Φm(T) in the algebraic closure of F are precisely the primitive mth roots of unity there, and that all these roots are simple.
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