Local Class Field Theory

Abstract

In this chapter we consider local fields, i.e., fields which are complete with respect to a discrete valuation and have finite residue class fields. The local fields are the p-adic number fields, i.e. the finite extensions K of the field k = Q _p of p-adic numbers (case char(K) = 0), and the finite extensions K of the power series field k=F _p ((x)) (case char (K) = p > 0). Here the module A _K of the abstract theory will be the multiplicative group K* of K. We therefore have to study the structure of this group. We introduce the following notation. Let

• v_K be the discrete valuation of K, normalized by v_K(K*) = ℤ,

• ϑK= aεK❘v_K(a)≥0 the valuation ring,

• p _K = aε _K \v _K (a)>0 the maximal ideal,

• K = ϑ _K /p _K the residue class field, and p its characteristic,

• U _K =aεK❘v _K (a) = 0 the group of units,

• U _k ⁽ⁿ⁾ = 1+p _{n K} the groups of higher principal units, n=l,2,…,

• q = q _k =*k

• ❘a❘_p = q^{−VK(a) the absolute value of aεK*}

• μ_n the group of n-th roots of unity, and μ_n(K) = μ _n ∩K*. By π _x , or simply π, we always mean a prime element of ϑ _K , i.e. p _K = πϑ _K , and we set (π) = π^k\kεℤ for the infinite cyclic subgroup of K* generated by π.