ABSTRACT
This chapter gives a brief overview of various aspects of the theory of non-Archimedean local fields. If a field k is a subfield of a field K, thenK is called an extension of k. The class of Galois extension is important since for them there exists a one-to-one correspondence between intermediate fields and subgroups of the Galois group. On the other hand, the Galois group is a natural “symmetry group” for analytic objects related to an extension. An important class of distributions consists of Borel measures on K, which are finite on any compact subset. The chapter focuses on the computation of some integrals over a local field K or its subsets, which contain expressions like X, where X is an additive character. Such integrals often appear in various problems of p-adic quantum mechanics, and spectral theory. The idea of the adeles is to incorporate in a single algebraic structure all completions of the field Q of rational numbers.
