Complete discrete valuation fields. Abelian local class field theories

Abstract

This chapter discusses local fields. Local behavior of a 1-dimensional scheme X near a “nice” point x is described by the local ring, whose completion is a complete discrete valuation ring with residue field k ( x ). The class of complete discrete valuation fields is closely connected with global fields—algebraic number and rational function fields. Local class field theory is one of the highest tops of classical algebraic number theory. It establishes a 1–1 correspondence between abelian extensions of a complete discrete valuation field F whose residue field is finite and subgroups in the multiplicative group F *. In the equal-characteristic cases for an arbitrary field K , there exists a complete discrete valuation field F , whose residue field is isomorphic to K . For the unequal-characteristic case: if K is a field of characteristic p , then there is a complete discrete valuation field F of characteristic 0 with prime element p and residue field K .