Criterion for the equality of norm groups of idele groups of algebraic number fields

[摘要] One of the fundamental theorems of global class field theory states that there is a one-to-one correspondence between finite Abelian extensions of an algebraic number field k and the norm groups of the idele class group C-k = J(k)/k* or k. More generally, for finite extensions K and L of k there is the following group theoretic interpretation of NK/kCK subset of or equal to NL/kCL. Let E be a finite Galois extension of k containing K and L, and let G = G(E/k), H = G(E/K), and N = G(E/L) be the corresponding galois groups. It follows by global class field theory that NK/kCK subset of or equal to NL/kCL iff G'H subset of or equal to G'N, where G' is the commutator subgroup of G. In the present work we prove that N(K/k)J(K) subset of or equal to N(L/k)J(L) iff every element of H of prime power order is conjugate in G to an element of N. We also show that the same group theoretic condition is equivalent to N(K/k)subset of or equal to N(L/k), where N(K/k) is the group of elements of k* that are local norms everywhere from K to k. We then use this group theoretic criterion to investigate the equality of norm groups as subgroups of k*. (C) 1997 Academic Press.

[发布日期] 1997-02-01