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

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.

• The different ideal appears in the functional equation of the Dedekind zeta function of a number field $K/\mathbb{Q}$. With the embeddings $\sigma_j : K \to \mathbb{C}$ see $\mathcal{O}_K$ as a lattice in $\mathbb{C}^n$, then the different is the dual lattice. Tate reformulated the Dedekind zeta function in adelic terms : with p-adic integrals, Fourier analysis, Schwartz functions and Poisson summation formula in the adele ring $\mathbb{A}_K$. Commented Aug 15, 2017 at 21:42
• @reuns I understand the trace and different ideal of a number field $K/\mathbb{Q}$. Here, I am asking about the trace and different of $k/\mathbb{Q}_p$. To clarify, $k=K_{\mathfrak{p}}$ is the $\mathfrak{p}$-adic extension of $\mathbb{Q}_p$. Commented Aug 15, 2017 at 22:25
• I see... And therefore I believe the different ideal would be $\{x\in K_{\mathfrak{p}} : Tr_{K_{\mathfrak{p}/Q_{p}}}(xy)\in Q_{p} \forall y\in K_{\mathfrak{p}}\}$. Commented Aug 15, 2017 at 22:30
• If $K/\mathbb{Q}$ is Galois, $\sigma \in Gal(K/\mathbb{Q}), \sigma(\mathfrak{p}) = \mathfrak{p}$ then $\sigma$ is well-defined on $\mathcal{O}_K/\mathfrak{p}^m$ as well as $\varprojlim_m \mathcal{O}_K/\mathfrak{p}^m$ and $K_\mathfrak{p}$, thus $$Tr_{K_\mathfrak{p}/\mathbb{Q}_p}(\alpha)=\frac{1}{|D_\mathfrak{p}/I_\mathfrak{p}|}\sum_{\sigma \in D_\mathfrak{p}}\sigma(\alpha)= \sum_{\sigma \in D_\mathfrak{p}/I_\mathfrak{p}}\sigma(\alpha)=\sum_{l= 1}^{f_p} Frob_\mathfrak{p}^l(\alpha)$$ Commented Aug 16, 2017 at 0:18

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.