Artin L-Functions and Galois Module Structure

Abstract

In Chap. 2 we explained class field theory as a theory connecting the abelian extensions of an algebraic number field K with the closed subgroups of finite index of the idele class group of K. A direct generalization of class field theory should consist of a topological group $\mathfrak{G}(K)$ , generalizing the idele class group, defined in terms of K with functorial properties with respect to field homomorphisms K→L and a canonical homomorphism ${\varphi_K}:\mathfrak{G}(K) \to G(\bar{K}/K): = {G_K}$ such that ϕ _K respects functorial behavior of $\mathfrak{G}(K)$ and G _K in the sense of Example 12 of Chap. 3 and such that U → ϕ _K (U) is a one to one correspondence between closed subgroups of finite index in $\mathfrak{G}(K)$ and closed subgroups of finite index in G _K . This last property can also be expressed saying that the induced homomorphism ${\hat{\varphi }_K}$ of the total completion (Chap. 3.1.1) of $\mathfrak{G}(K)$ into G _K is an isomorphism onto G _K .