Local Class Field Theory

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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 vK be the discrete valuation of K, normalized by vK(K*) = ℤ, ϑK= aεK❘vK(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* μn the group of n-th roots of unity, and μn(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) = πk\kεℤ for the infinite cyclic subgroup of K* generated by π. KeywordsGalois GroupPrime ElementAbelian ExtensionClass Field TheoryHilbert SymbolThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.