Scholz admissible moduli of finite Galois extensions of algebraic number fields

概要

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Let K be a finite Galois extension over an algebraic number field k with Galois group G. We call a modulus m of K Scholz admissible when the Schur multiplier of G is isomorphic to the number knot of K/k modulo m. This paper develops a systematic treatment for Scholz admissibility. We first reduce the problem to the local case, in particular, to the strongly rami-fied case, and study this case in detail. A main object of local Scholz admissi-bility is H^<-1>(G,U^(s)_k) in the strongly ramified case. In the case where K/k is totally strongly ramified of prime power degree p^n, we prove that the natural homomorphism: H^<-1>(G,U^(r+s)_k)→H^<-1>(G,U^(s)_k) is trivial for s ≥ 1, where r denotes the last ramification number, This result describes a basic situation for vanishing of H^<-1>(G,U^(s)_k). Using this result for a Galois tower K⊃L⊃k with a totally strongly ramified cyclic extension L/k we prove a relationship between Scholz admissible moduli of K/L and K/k. This gives a way to estimate for Scholz conductor of K/k from the ramification in K/k. As an application of this result we give an alternative proof of a result of Frohlich.
新潟大学の論文
2003-00-00