Math 621  Problem Set #6  Due Fri., April 5, 2002

1. Let K be a number field, and let f(x) ∈ K[x] be a polynomial with distinct roots (in ℂ). Suppose that f splits into a product of linear factors modulo 𝔭, for all but finitely many primes 𝔭 ⊂ O_{K}. Prove that f splits into a product of linear factors in K[x]. [Hint: Let E be the splitting field of f. Which primes of K split in L?]

2. For any finite extension F ⊃ E of number fields (not necessarily Galois), let S_{F/E} be the set of primes P of F that are of degree 1 over K (i.e. such that e_{P} = f_{P} = 1), and let Spl′_{F/E} be the set of primes 𝔭 of E over which there is a prime P of F that lies in S_{F/E}.

a) Show that S_{F/E} has Dirichlet density equal to 1. [Hint: PS5 #3b.]

b) Let E, L be number fields, with E Galois over ℚ. Suppose that Spl′_{L/ℚ} ≺ Spl_{E/ℚ}. Prove that E ⊂ L. [Hint: How do the primes in S_{L/ℚ} behave in the compositum EL? Now apply (a) to the extension L ⊂ EL.] Compare and contrast with PS5 #5b.

c) Let L be a number field. Show that the following are equivalent:

(i) L is Galois over ℚ.

(ii) Every prime number p that has a factor of degree 1 in L actually splits completely in L.

[Hint: For (ii) ⇒ (i), apply PS5 #5a and then (b) above with E = the Galois closure of L over ℚ.]

3. Find all number fields K such that every prime number p ≡ 1 (mod 12) splits completely in K.

4. Let χ : (ℤ/m)^{∗} → ℂ^{∗} be a non-trivial homomorphism. Consider the sum ∑_{n} χ(n)/n^{s}, where the index n runs over positive integers that are relatively prime to m. Show that the sum converges for Re s > 0, and uniformly on {s ∈ ℂ | Re s > δ, |Im s/Re s| ≤ c} for any given δ, c > 0; and hence it converges at s = 1. [Hint: For each N, let F (N) = ∑ χ(n), where n ranges over integers prime to m in the range 1 ≤ n ≤ N. Then F (N) is a bounded function of N.]

5. a) Which of the following fields are contained in a cyclotomic field ℚ(ζ_{m})? For those that are, find the smallest such m: ℚ(\sqrt{37}), ℚ(\sqrt{19}), ℚ( ^{3} \sqrt{7}), ℚ(cos \frac{π}{10}).

b) Find an abelian extension L of K = ℚ(\sqrt{2}) that does not lie in ℚ(ζ_{m}) for any m. [Hint: Let L = K(\sqrt{α}), where α is chosen so that L is not Galois over ℚ. If L ⊂ ℚ(ζ_{m}), is Gal(ℚ(ζ_{m})/L) a normal subgroup of Gal(ℚ(ζ_{m})/ℚ)?]

6. Find the solutions to the Diophantine equation

x^{3} + 2y^{3} + 4z^{3} − 6xyz = ±1.

[Hint: Units in ℤ[ ^{3} \sqrt{2}].]