Local Class Field Theory

Abstract

and F. K. Schmidt [11], establishes a 1-1 correspondence between the normal abelian extensions L/K and the factor groups K*/H, where K* is the multiplicative group of K and H is a subgroup of finite index in K*. This correspondence is such that H coincides with the multiplicative group NLK(L*) of the norms of the elements of L*, and there is established an between the Galois group of L/K and the group K*/H. This isomorphism is usually defined by means of the residue which maps K* onto the Galois group of L/K. Originally, the local class field theory was derived from the much more difficult class field theory 'in the large', for algebraic number fields. Chevalley has given an independent derivation of the local theory in [3]. These results were also obtained (but not published), at the same time, by F. K. Schmidt. Perhaps the simplest known exposition is an alternative version given by Chevalley in [4] which makes use of the theory of simple non-commutative algebras in establishing the fundamental properties of the norm residue symbol. The exposition which we give here differs from that of Chevalley in two respects: First, the rather indirect definition of the fundamental by the norm residue symbol (which is defined only for cyclic extensions to begin with and then has to be 'pieced together') is replaced by that of a more transparent and more general inverse mapping which is valid also for the non-abelian case and is proved to be an in the abelian case. This mapping is defined by means of a homomorphism involving factor sets which was discovered by Akizuki and Nakayama in 1936, [1] and [9]. Using the properties of the normresidue symbol, Nakayama has proved that this yields the inverse to the fundamental isomorphism. We shall, however, dispense with the norm residue symbol altogether and prove directly certain properties of the factor set homomorphism which are sufficient to establish the local class field theory. The second departure from Chevalley's procedure is that the use of noncommutative algebras is eliminated. This is possible by developing the theory of factor sets in an intrinsic manner within the frame work of the EilenbergMacLane cohomology theory for groups, [7], [8]. It is hoped that our procedure makes the local class field theory more easily accessible. Our proofs are elementary, in the sense that-except for the later use of the arithmetic results enumerated in ?5-they require only general field