Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Lecture 20: Ramification in Number Fields October 12, 2001

Reference. Neukirch, Section 2.8, and to a lesser extent Section 1.8.

Outline of lectures.

1. (Left over from last time) Work out the structure of ℚ_{p}(ζ_{p^{n}}). In particular, show that 1 − ζ_{p^{n}} generates the valuation ring.

2. Let L/K be an extension of number fields, let 𝔮 be a nonzero prime ideal of 𝔬_{L}, and let 𝔭 = K ∩ 𝔮. As noted earlier, the completion L_{𝔮} is a finite extension of K_{𝔭}. Define the ramification index and inertia degree e(𝔮/𝔭) = e(L_{𝔮}/K_{𝔭}) and f(𝔮/𝔭) = f(L_{𝔮}/K_{𝔭}).

3. Prove that the 𝔭-adic valuation does not necessarily extend uniquely to L, in contrast to the case where K was a local field. Rather, the extensions are precisely the 𝔮-adic valuations for all primes 𝔮 containing 𝔬_{L}𝔭, or equivalently, all primes of 𝔬_{L} in the prime factorization of 𝔬_{L}𝔭. (The point: given an extension of the valuation to L, the maximal ideal of the valuation ring is a prime ideal containing 𝔬_{L}𝔭.)

4. For L/K a finite extension of number fields and 𝔭 a prime of 𝔬_{K}, prove the fundamental

identity

[L : K] = ∑_{𝔮}[L_{𝔮} : K_{𝔭}] = ∑_{𝔮}e(𝔮/𝔭)f(𝔮/𝔭),

where the sum runs over all primes 𝔮 of 𝔬_{L} containing 𝔭.

5. Point out another interpretation: the tensor product L ⊗_{K} K_{𝔭} factors as the product of the L_{𝔮}.

6. Prove that 𝔬_{L}𝔭 = ∏_{𝔮} 𝔮^{e(𝔮/𝔭)}. That is, the ramification index is the factor to which 𝔮 shows up in the prime factorization of 𝔬_{L}𝔭.

7. Discuss some examples, e.g., ℚ(i) and/or ℚ(x)/(x^{3} + x + 1). Point out (to be proved later) that when K = ℚ, the only primes of K over which ramification happens are the ones dividing the discriminant of L. (One part of this claim is on this week’s homework.) Later we will define the “relative discriminant”, which will allow to make a similar statement for general K.