Math 621  Problem Set #5  Due Wed., March 20, 2002

1. a) Redo Problem Set 4, problem 3, with ζ_{5} replaced by ζ_{m}, for an arbitrary positive integer m = ∏_{i=1}^{r} p_{i}^{α_{i}}.

b) For which m is there a prime p that remains prime in ℤ[ζ_{m}]? [Hint: What is the structure of Gal(ℚ(ζ_{m})/ℚ), and what is the order of Frobenius of such a p?]

c) For which m is there a prime p that is totally ramified in ℤ[ζ_{m}]?

d) How many quadratic extensions of ℚ are contained in ℚ(ζ_{m})? [Hint: First do the case r = 1.]

2. Show that if a set M of rational primes has a well-defined probabilistic density

lim_{n→∞} #{p ∈ M | p ≤ n}/π(n),

then it has a well-defined Dirichlet density

lim_{s→1^{+}} ( ∑_{p∈M} p^{−s})/log(\frac{1}{s − 1}),

and that the densities are equal.

3. Verify that if K is a number field and N𝔭 denotes the norm of a prime 𝔭 of O_{K}, then:

a) For any δ > 0, ∑_{𝔭} ∑_{n=2}^{∞} \frac{1}{nN𝔭^{ns}} converges uniformly and absolutely on Re s ≥ \frac{1}{2} + δ

and so is holomorphic in a neighborhood of s = 1.

b) ∑_{{𝔭 s.t. \atop f𝔭>1}} \frac{1}{N𝔭^{s}} is holomorphic near = 1. [Hint: Write the sum as ∑_{p} ∑_{{𝔭|p s.t. \atop f𝔭>1}} \frac{1}{p^{sf_{𝔭}}}, and

use that ∑ \frac{1}{p^{2}} converges.]

4. Verify the Artin Reciprocity law (i.e. that σ : I(c)/P_{c}N (c) →^{∼} Gal(L/K)) for the case that K = ℚ, L = ℚ(\sqrt{p}) (where p is an odd prime), with c = pv_{∞} if p ≡ 1 (mod 4), and with c = 4pv_{∞} if p ≡ −1 (mod 4).

5. a) Let K ⊂ L be a finite extension of number fields, and let L̃ be the Galois closure of L over K. Show that Spl_{L/K} = Spl_{L̃/K}. [Hint: Let 𝔭 be a prime of K, and let P be a prime of L over 𝔭. Let K_{𝔭} be the 𝔭-adic completion of K, and let K̄_{𝔭} be its algebraic closure. Pick an embedding α : L ↪ K̄_{𝔭} as K-algebras. Show that the condition that a prime 𝔭 is in either Spl is equivalent to saying that every conjugate of L has the property that its image under α is contained in K_{𝔭}.]

b) Consider the result that if L, E are finite Galois extensions of a number field K, then

Spl_{L/K} ≺ Spl_{E/K} ⇔ E ⊂ L.  (∗)

Does this assertion remain true if L is permitted not to be Galois? If E is permitted not to be Galois? If both are permitted not to be Galois? In each case, either give a proof of (∗) or a counterexample, in the latter event explain where the proof in the Galois case breaks down in the more general situation.