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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