Cohomology of Local Fields

Abstract

AbstractWe now begin the development of cohomology in number theory. As a ground field we take a nonarchimedean local field k, i.e. a field which is complete with respect to a discrete valuation and has a finite residue field. This covers two cases, namely p-adic local fields, i.e. finite extensions of \(\mathbb{Q}_{p}\) for some prime number p, and fields of formal Laurent series \(\mathbb{F}((t))\) in one variable over a finite field. For the basic properties of local fields we refer to [160], chapters II and V. As always, \(\bar{k}|k\) denotes a separable closure of k and K|k the subextensions of \(\bar{k}|k\). v k denotes the valuation of k, normalized by \(v_{k}(k^{\times}) = \mathbb{Z}\), and κ the residue field. For every Galois extension K|k we set $$H^i(K|k) := H^i(G(K|k),K^{\times}),\quad i \geq 0. $$ If K|k is finite, we also set \(\hat{H}^{i}(K|k) = \hat{H}^{i}(G(K|k),K^{\times})\) for \(i \in \mathbb{Z}\). The basis for the results in this chapter is the following theorem.Mathematics Subject Classification11Gxx11Rxx11Sxx12Gxx14Hxx20Jxx