Class numbers of algebraic number fields of Eisenstein type, II

Abstract

Let K be an algebraic number field, of degree n, with a completely ramifying prime p, and let t be a common divisor of n and (p − 1) 2 . Then it is proved that if K does not contain the unique subfield, of degree t, of the p-th cyclotomic number field, then we have ( h K , n) > 1, where h K is the class number of K. As applications, we give several results on h K of such algebraic number fields K.