New Characterizations of Kronecker Equivalence

AbstractTwo number fields K|k, K′|k are called Kronecker equivalent over k iff the sets of primes of k having a prime divisor of first degree in K respectively in K′ coincide up to at most finitely many exceptions. We give some new characterizations of Kronecker equivalence, which rely on a refinement (Theorem 3) of Klingen′s explicit calculation of MU(FN|k(P)) and on a result of Artin on representations of finite groups. Further we show how the Dirichlet series expansions of the Dedekind zeta functions of Kronecker equivalent fields are related. Moreover, our characterization gives information on the kernel of Artin L-functions and on class groups of Kronecker equivalent fields. Using character relations we define Kronecker equivalence and arithmetical equivalence for tupels of fields. This leads to a generalization of Artin′s theorem that over Q relations for zeta functions and character relations are the same to arbitrary ground fields; one only has to replace relations for zeta functions by some combinatorical conditions on residue degrees (Theorem 17)