Tate's thesis: Trace function & different ideal of $\mathfrak{p}$-adic fields

Question

I have started reading Tate's thesis. $k$ is a completion of an algebraic number field at a prime divisor $\mathfrak{p}$. Let $\mathfrak{p}$ divide the rational prime $p$. I am unable to understand what the absolute different ideal of $k$ would be. Even before that, I don't understand what the trace function of k over $\mathbb{Q}_{p}$ would be. Any source and explicit examples would be of great help.

Answer 1

First, one should have in mind that for a finite-degree separable field extension $L/K$, the trace pairing $\langle x,y\rangle={\mathrm tr}^L_K xy$ is non-degenerate. Here the trace is Galois trace, which, notably, does not depend on the extension being Galois. Rather, ${\mathrm tr}x$ is the sum, in a Galois closure of $L$ over $K$, of all the conjugates of $x$... which, by Galois theory, lies in the ground field $K$, in fact.

Thus, for example, for a finite (unavoidably separable, because characteristic is $0$) extension $k/\mathbb Q_p$, trace is non-degenerate, and is "sum of conjugates". An often unasked question amounts to comparison of the "global trace" (of number field $K/\mathbb Q$) to the local traces $K_v/\mathbb Q_p$, where $v$ runs through places/primes of $K$ lying over $p$. A very useful point, too-often mistakenly suppressed because it's "not sufficiently elementary", is that $K\otimes_{\mathbb Q} \mathbb Q_p\approx \bigoplus_{v/p} K_v$. In this context, it is not at all surprising that "the global trace is the sum of the local traces".

And, yes, for number field extensions $K/k$ and for local field extensions $K/k$, the inverse different is the fractional-ideal inverse of $\{x\in K: {\mathrm tr}x\mathfrak o_K\subset \mathfrak o_k\}$. In both cases, in addition to giving a duality by the trace pairing, there are strong connections to ramification.

(Also, I'd suggest that K. Iwasawa be mentioned in this context, since, after all, he gave an ICM talk in 1950 about doing this sort of thing. Not to mention Matchett's 1946 thesis under Artin about similar matters. "It was in the air.")