The abstract class field theory that we have developed in the last chapter is now going to be applied to the case of a local field, i.e., to a field which is complete with respect to a discrete valuation, and which has a finite residue class field. By chap. II, (5.2), these are precisely the finite extensions K of the fields ℚ p or F p ((t)). We will use the following notation. Let υ K be the discrete valuation normalized by υ K (K*) = ℤ, O K = {a ∈ K | υ K (a) ≥ 0} the valuation ring, p K = {a ∈ K | υ K (a) > 0} the maximal ideal, κ = O K /p K the residue class field, U K = {a ∈ K* | υ K (a) = 0} the unit group, U K (n) = 1 + p K n the group of n-th higher units, n = 1, 2, ... , q = q K = #κ, |a|p = q −υ K (a) the normalized p-adic absolute value, μ n the group of n-th roots of unity, and μ n (K) = μ n ∩ K*. π K , or simply π, denotes a prime element of K, i.e., p K = πO K .