Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Lectures 15-16: Completions and Local Fields October 1 and 3, 2001

Reference. Neukirch, Sections 2.4 and 2.5, and a bit of 2.6.

Jargon watch. An exponential valuation is discrete if it takes values in a discrete subgroup of ℝ, which can always be normalized to be ℤ itself. A valuation is discrete if it is nonarchimedean and comes from a discrete exponential valuation. (For example, the p-adic valuation on ℚ is discrete.)

Recall that given a field with a nonarchimedean valuation, the valuation ring of the field is the set of elements of norm at most 1.

Outline of lectures.

1. For K a number field and 𝔭 a nonzero prime ideal, define K_{𝔭} as the completion of K with respect to the valuation | · |_{𝔭}. If 𝔭 ∩ ℚ = (p), then [K_{𝔭} : ℚ_{p}] ≤ [K : ℚ], but the inequality can be strict.

2. Demonstrate some examples for the Gaussian numbers ℚ(i), with 𝔭 = (2 + i), 𝔭 = (3), and the “anomalous” case 𝔭 = (1 + i).

3. Define a local field as a field which is complete with respect to a discrete valuation, with finite residue field. Note: the only such fields are finite extensions of ℚ_{p} or of the power series field 𝔽_{p}((t)).

4. Focus a bit on discrete valuation rings. (Remember, these aren’t necessarily complete.) Note that the maximal ideal 𝔪 of a DVR is generated by any element of 𝔪 \ 𝔪^{2}, and every ideal of a DVR is a power of the maximal ideal. In particular, a DVR is a principal ideal domain. For U_{i} the group of units congruent to 1 modulo π^{i}, point out the isomorphisms U_{0}/U_{1} ≅ (R/𝔪)^{∗} and U_{i}/U_{i+1} ≅ R/𝔪 for i ≥ 1.

5. State and prove Hensel’s lemma: if a monic polynomial over the ring of integers in a local field factors, modulo the maximal ideal, into two coprime factors, then there is a corresponding factorization of the original polynomial. Example from last time: if n is a quadratic residue modulo p, then x^{2} − n factors over ℤ_{p}.

6. State the extension theorem: if L/K is a finite extension of fields, and K is complete with respect to the valuation | · |, then there exists a unique extension of | · | to L. Sketch the proof in the nonarchimedean case (the archimedean case uses Ostrowski’s theorem, see Neukirch 2.4).

7. Compare the previous theorem to the case ℚ(i)/ℚ of an extension of number fields: a valuation downstairs may extend upstairs in multiple ways. Sneak preview: if K is a number field and 𝔭 is a prime ideal of 𝔬_{K}, the extensions of | · |_{𝔭} to a finite extension L of K correspond to the prime factors of 𝔭𝔬_{L}.

8. Define the Newton polygon of a polynomial over a discrete valuation ring. Show that the slopes of the Newton polygon coincide with the valuations of the roots of the polynomial.

9. If R is the valuation ring of a local field, observe that any irreducible polynomial over R only has a single slope in its Newton polygon (by the extension theorem).