Math 514a Algebraic Number Theory Fall 1995

Homework assignment 4:

18. Write ζ_{n} for a primitive n-th root of unity.

a) Show that if p is an odd prime, Q(ζ_{p}) contains a unique quadratic field. Which one is it?

b) Show that every quadratic field is contained in a cyclotomic field Q(ζ_{n}). (This is a special case of the famous Kronecker-Weber theorem, according to which every Galois extension of Q with an abelian Galois group is contained in some Q(ζ_{n}).

19. Notation as in the last problem. Let K ⊆ Q(ζ_{n}) be a number field.

a) Let H ⊆ G = Gal(Q(ζ_{n})/Q) be the group corresponding to K by Galois theory. Let σ_{1}, …, σ_{d} (d = [K : Q]) be coset representatives for H in G. Set

η_{i} = ∑_{τ∈H} ζ_{n}^{σ_{i}τ} .

(These are the “Gaussian periods.”) Show that any one of the η_{i} generates K over Q.

b) Show that if n is a prime then η_{1}, …, η_{d} is a basis of the group O_{K}. Is this true for general n?

c) Is it true (say for n prime) that O_{K} = Z[η_{i}] for some i?

20. Here is a large supply of non-PID integer rings: Let K = Q(\sqrt{−p}) where p is a prime ≡ 1 (mod 4). Show that the ideal (2, 1 + \sqrt{−p}) has norm 2 but no element of K has norm 2 and so O_{K} is not a PID. Can you generalize this example?

21. Let K be a number field. For each maximal ideal ℘ ⊆ O_{K}, and each x ∈ K^{×}, let v_{℘}(x) be the exponent of ℘ in the factorization of the principal fractional ideal (x) and let |x|_{℘} = p^{−v_{℘}(x)} where p is the rational prime under ℘. By convention we set v_{℘}(0) = +∞ and |0|_{℘} = 0.

a) Prove that | · |_{℘} is a metric on K, i.e., i) |x|_{℘} = 0 ⇔ x = 0 ii) |xy|_{℘} = |x|_{℘}|y|_{℘} iii) |x + y|_{℘} ≤ |x|_{℘} + |y|_{℘}. In fact, |x + y|_{℘} ≤ max(|x|_{℘}, |y|_{℘}) with equality unless |x|_{℘} = |y|_{℘}.

b) For any subset M of the maximal ideals of O_{K}, let R = {x ∈ K| |x|_{℘} ≤ 1 for all ℘ ∈ M}. Prove that R is a subring of K containing O_{K}. Describe explicitly the cases where M is empty, M consists of a single ideal, and M is the set of all maximal ideals.

22. Let K be a number field such that O_{K} is a PID. Give a complete classification of all rings R with O_{K} ⊆ R ⊆ K. (Something to think about: what happens for a general number field?)

23. Suppose that f(X) = X^{n} + a_{1}X^{n−1} + ⋯ + a_{n} ∈ Z[X] is an irreducible Eisenstein polynomial for the prime p, i.e., p|a_{i} for all i and p^{2} |̸a_{n}. Let K be the number field generated by a root of f.

a) Show that p ramifies totally in O_{K}, i.e., there is a unique prime ℘ ⊆ O_{K} over p and (p) = ℘^{n}.

b) Find a suitable f so that K = Q(ζ_{p^{m}}) and identify the ideal ℘ explicitly in this case.

24. Show that the different is multiplicative in towers. (Marcus, exercise 38, p. 96.)

25. Figure out as much as you can about factorization in your favorite type of field. For biquadratics, you should be able to give a fairly complete answer (based on congruences). For pure cubics, the situation is more complicated. See Marcus, exercise 26, p. 89 for some ideas. It may also be interesting to look at the fields Q(ζ_{3}, ^{3} \sqrt{a}).