Abelian extensions of arbitrary fields

Abstract

For a field, a normal extension of is a field F containing such that the group of automorphisms of F leaving point-wise fixed (the Galois group) is finite and leaves no more than fixed. It is an untouched classical problem to determine the normal extensions of k. Because of this, realistic work has centered on finding the Abelian extensions (the normal extensions where the Galois group is commutative). Where is an algebraic number field this makes up the class field theory. Where is any field of characteristic zero containing all the roots of unity, the Abelian extensions are given by the Kummer theory. In this paper we generalize the Kummer theory to an arbitrary field. In the characteristic p case or in the case where roots of unity do not exist, our answer, although it does not involve field extensions and thus is technically correct, is not as explicit as we could wish. For instance, it is not clear how to use this work in the derivation of the class field theory. Yet our answer is in terms of a cohomology theory in which a great deal of machinery exists simply because the theory is exactly analogous to (i.e., is the same in category theory as) the cohomology of groups, and for this reason we feel it presents a natural, systematic approach to questions involving Abelian extensions in the same way that the less general Kummer theory provides such an approach. If we replace the word set by commutative with identity over k and the phrase map from A to by algebra homomorphism from B to A, then the concept of a group transforms (using category theory for precision) to an object which is often called a group scheme (or equivalently, a Hopf with inverse map). The ordinary cohomology of groups, together with all its formalistic properties, transforms to these schemes. The group rings k(H) and k(L) are such schemes, where H denotes the group of integers and L denotes the rationals modulo the integers. Our result is that the second cohomology group of k(L) with coefficients in k(H) (trivial operation) is naturally isomorphic to the character group of the full Galois group of k. This means that the Abelian extensions of are in one-one correspondence with the finite subgroups of of H2(k(L), k(H)). This cohomology group is explicitly a certain factor group of