Math 621  Problem Set #4  Due Mon., March 4, 2002

1. a) Find the different and discriminant of ℤ[\sqrt{5}] and ℤ[\frac{1+\sqrt{5}}{2}] over ℤ. What is the relationship between these rings, and how do the different and discriminant of one ring compare with those of the other?

b) Do the same for ℂ[x, y]/(y^{2} − x(x − 1)^{2}) and ℂ[x, z]/(z^{2} − x) over ℂ[x]. Also give a geometric interpretation.

2. Let A = ℤ_{3} and B = A[\sqrt{−1}, \sqrt{3}]. Is there an element β ∈ B such that B = A[β]? If so, find such a β.

3. For each prime p of ℤ, describe the factorization of p in ℤ[ζ_{5}], where ζ_{5} is a primitive fifth root of unity. In particular, find e, f, r, find the inertia and decomposition groups, and find the Frobenius automorphisms.

4. a) Suppose that K is a Galois extension of ℚ with Galois group G = ℤ/4. What are the possibilities for the decomposition group G_{p} and inertia group T_{p}, as p varies? Illustrate by examples. Also, what is the order of Frobenius in G for a prime p in ℤ that remains prime in O_{K}?

b) Redo part (a) with G = ℤ/4 replaced by H = ℤ/2 × ℤ/2.

5. a) Let A ⊂ B be an extension of complete discrete valuation rings whose residue fields have characteristic p. Let v, w be the (additive) valuations at the maximal ideals 𝔭, P of A, B respectively, and let e be the ramification index. Show that the different satisfies D ⊃ P^{c}, where c = e − 1 + w(e). [Hint: Use the same hint as in PS 3 #4, using the Eisenstein polynomial f of the uniformizer π of B. Show that the non-zero terms of f′(π) have the property that their values under w are distinct (even mod e), so that w(f′(π)) is the minimum of these values; then consider the contribution from the highest degree term of f′.]

b) Combining this with PS 3 #4, show that there are upper and lower bounds for the exponent of P in D, if A is the ring of integers of a p-adic field (i.e. a finite extension of ℚ_{p}).

c) Why does (a) not give an upper bound if A = k[[x]], where k is a finite field? Is there any upper bound in terms of e? [Hint: Consider the discriminant of the extension given by y^{p} − x^{n(p−1)}y = x.]

6. Let M be the set of p-adic absolute values on ℚ (including the “usual” absolute value |·|_{∞}). Let ℚ_{p} be the corresponding completion (so that ℚ_{∞} = ℝ).

a) Let p_{1}, …, p_{n} ∈ M be distinct, and let α_{i} ∈ ℚ_{p_{i}} for i = 1, …, n. Show that there is an a ∈ ℚ such that |α_{i} − a|_{p_{i}} ≤ 1 for all i and |a|_{p} ≤ 1 for all p ≠ p_{1}, …, p_{n}. [Hint: For each p_{i} that is finite, p_{i}-adically approximate α_{i} by an element a_{i} ∈ ℤ[1/p_{i}]. Then add an appropriate integer to ∑ a_{i} to handle the infinite prime.]

b) Let p_{0}, p_{1}, …, p_{n} ∈ M be distinct, and let ε > 0. Show that there is a non-zero rational number c such that |c|_{p_{i}} < ε for i = 1, …, n and such that |c|_{p} ≤ 1 for all p ≠ p_{0}, p_{1}, …, p_{n}. [Hint: First do this in the case where p_{0} = ∞; then adjust your result to handle the case that p_{0} is finite.]

c) Let p_{0}, p_{1}, …, p_{n} be distinct elements of M, and let α_{i} ∈ ℚ_{p_{i}} for i = 1, …, n. Let ε_{1}, …, ε_{n} > 0. Show that there exists exists α ∈ ℚ such that |α − α_{i}|_{p_{i}} < ε_{i} for i = 1, …, n and such that |α|_{p} ≤ 1 for all p ≠ p_{0}, p_{1}, …, p_{n}. [Hint: Apply part (b) to ε = min(ε_{i}) and obtain c ∈ ℚ; now apply part (a) to α_{i}/c to obtain an a; then consider ca.]

d) Show that part (c) includes PS1 #1(a) and the Chinese Remainder Theorem for ℤ as special cases.