Criterion for the Equality of Norm Groups of Idele Groups of Algebraic Number Fields Original Research Article

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 fieldkand the norm groups of the idele class groupCk=Jk/k* ofk. More generally, for finite extensionsKandLofkthere is the following group theoretic interpretation ofNK/kCKsubset of or equal toNL/kCL. LetEbe a finite Galois extension ofkcontainingKandL, and letG=G(E/k),H=G(E/K), andN=G(E/L) be the corresponding Galois groups. It follows by global class field theory thatNK/kCKsubset of or equal toNL/kCLiffG′Hsubset of or equal toG′N, whereG′ is the commutator subgroup ofG. In the present work we prove thatNK/kJKsubset of or equal toNL/kJLiff every element ofHof prime power order is conjugate inGto an element ofN. We also show that the same group theoretic condition is equivalent toN(K/k)subset of or equal toN(L/k), whereN(K/k) is the group of elements ofk* that are local norms everywhere fromKtok. We then use this group theoretic criterion to investigate the equality of norm groups as subgroups ofk*.