Integral representations afforded by ambiguous ideals in some abelian extensions

Abstract

Let K F be a finite cyclic extension with Galois group π of order m, F the quotient field of a Dedekind domain R, Λ the maximal order of Fπ. We study the Rπ-representations afforded by ambiguous ideals U of K. Suppose K resembles the prime power degree subfields of Q( l √1), l prime; then we have an explicit description of Λ ⊗ Rπ U in terms of Λ ⊗ Rπ O K and the ramification of K F . The key idea is to form the Kummer extension Kk k , k = F( m √1) , again unramified at all primes dividing m. In the local case where k = Q p ( m √1), m = p n , we describe elements x of k such that the extension k( x 1/ m )/ k is unramified and cyclic of degree m.