Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Lectures 32-33: Different and Discriminant November 19-21, 2001

Reference. Neukirch, Section III.2.

Outline of lectures.

Unless otherwise specified, L/K will be a finite extension either of number fields or of finite extensions of ℚ_{p}.

1. Define the relative norm Norm_{L/K}(𝔞) of a fractional ideal 𝔞 of L as the fractional ideal of K generated by Norm_{L/K}(α) for all α ∈ 𝔞. Notice that for L/K an extension of number fields, the relative norm can be computed locally:

Norm_{L/K}(𝔞) = ∏_{𝔭} ∏_{𝔮}Norm_{L_{𝔮}/K_{𝔭}}(𝔞) = ∏_{𝔭} ∏_{𝔮}𝔭^{f(𝔮/𝔭)v_{𝔮}(𝔞)}.

2. For L/K any finite extension of fields, the trace pairing ⟨·, ·⟩ : L × L → K is given by ⟨x, y⟩ = Trace_{L/K}(xy). The trace pairing is nondegenerate (i.e., ⟨x, y⟩ = 0 for all y implies x = 0) if and only if L/K is separable.

3. Notice that the trace pairing takes 𝔬_{L} × 𝔬_{L} into 𝔬_{K}. (The trace of an algebraic integer is the sum of algebraic integers.)

4. Define the inverse different 𝔇_{L/K}^{−1} as the set of x ∈ L such that ⟨x, y⟩ ∈ 𝔬_{K} for all y ∈ 𝔬_{L}. Note that this is a fractional ideal of 𝔬_{L}.

5. Define the different 𝔇_{L/K} as the inverse of the inverse different; this is always an integral ideal. Note that 𝔇_{L/K} = ∏_{𝔭} ∏_{𝔮} 𝔇_{L_{𝔮}/K_{𝔭}}.

6. If M/L is another extension, note that 𝔇_{M/K} = 𝔇_{M/L}𝔇_{L/K}, as an ideal of 𝔬_{M}. The point is that Trace_{L/K}(Trace_{M/L}(x)) = Trace_{M/K}(x), so the trace pairing from M to K can be computed “all at once” or “in two stages”.

7. In case 𝔬_{L} = 𝔬_{K}[α] for some α ∈ 𝔬_{L}, let P (x) be the minimal polynomial of α over 𝔬_{L}. Show that 𝔇_{L/K} = (P′(α)) by showing that the dual basis of 1, α, …, α^{n−1} is

\frac{b_{0}}{P′(α)}, ⋯, \frac{b_{n−1}}{P′(α)},

where P (x)/(x − α) = b_{0} + b_{1}x + ⋯ + b_{n−1}x^{n−1}. In the general case, 𝔇_{L/K} is the ideal generated by P′(α) for all pairs α, P of an element α of 𝔬_{L} which generates L over K, and the minimal polynomial P of α.

8. Show that the relative discriminant is the relative norm of the different, by noting that Disc(1, α, …, α^{n−1}) = Norm_{L/K}(P′(α)). (For another approach, see Neukirch, Theorem III.2.9.)

9. Let 𝔮 be a prime of L over 𝔭 and let e = e(𝔮/𝔭) be the ramification index. Show that if 𝔮/𝔭 is unramified or tamely ramified, then v_{𝔮}(𝔇_{L/K}) = e − 1; otherwise, e ≤ v_{𝔮}(𝔇_{L/K}) ≤ e − 1 + v_{𝔮}(e). In particular, 𝔮 is ramified over 𝔭 if and only if 𝔮 divides 𝔇_{L/K}.