Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Problem Set 7: Local Fields/Ramification Due in Frank Calegari’s mailbox Monday, October 15

This week only: please make any extension requests directly to Frank. His email is

fcale@math.berkeley.edu.

Please submit seven of the following problems.

Jargon watch. Given a field K and a fixed exponential valuation on K, the value group is the image of the exponential valuation in ℝ.

A field with a discrete valuation is called henselian if the statement of Hensel’s Lemma holds over the field, whether or not the field is complete. For the purposes of this problem set, you may use the fact that a field K is henselian if and only if the following weak form of Hensel’s Lemma is true: if P (x) is a polynomial over the valuation ring R of K and α is a simple root of P over the residue field of R, then there is a root of P in R that reduces to α. See Section II.6 for more on henselian fields.

Throughout the set, let ζ_{n} denote an n-th root of unity.

More on local fields.

1. Determine all of the quadratic extensions of ℚ_{p}. (Your answer will have a different form for p = 2.)

2. Determine the degree of ℚ_{p}(ζ_{n}) over ℚ_{p}, for p a prime not dividing n.

3. Let K be the maximal unramified extension of ℚ_{p}. Prove that K is henselian. (Hint: reduce to a finite extension of ℚ_{p}.)

4. Let R be the ring of power series over ℂ that converge in some neighborhood of 0. Prove that the fraction field of R is henselian.

5. Neukirch II.6.1 (modified): Let f(x) = x^{n} +∑_{i=0}^{n−1} f_{i}x^{i} be a monic polynomial of degree n over a local field K, which factors completely over K with roots r_{1}, …, r_{n}. Prove that for any ϵ > 0, there exists δ > 0 such that if g(x) = x^{n} + ∑_{i=0}^{n−1} g_{i}x^{i} is a monic polynomial of degree n such that |f_{i} − g_{i}| < δ, then g has n roots s_{1}, …, s_{n} in K, which can be labeled so that |r_{i} − s_{i}| < ϵ for i = 1, …, n. That is, the roots of f vary continuously with the coefficients.

Ramification.

1. Prove that a finite extension of ℚ_{p} has only one unramified extension of any given degree. (Hint: recall that a finite field has only one extension of any given degree.)

2. Show that ℚ_{p}(ζ_{p}) = ℚ_{p}(π) for π a (p − 1)-st root of −p. (Hint: both extensions are tamely ramified.)

3. Neukirch II.7.1: The maximal unramified extension of ℚ_{p} is obtained by adjoining all roots of unity of order prime to p.

4. Neukirch II.7.3: Let K be a local field and L/K a totally and tamely ramified extension, and let Γ and Δ be the value groups of K and L, respectively. Show that the subfields of L containing K are in one-to-one correspondence with the subgroups of Δ/Γ.

5. Let P (x) be a monic irreducible polynomial over ℤ_{p}. Suppose the discriminant of P (x) is not divisible by p. Prove that the field ℚ_{p}[x]/(P (x)) is unramified over ℤ_{p}.

6. For p odd, prove that ℚ_{p}(ζ_{n}) is ramified over ℚ_{p} if and only if p divides n. Optional: what happens if p = 2? (Hint: see the end of Section II.7.)