PRINCIPALIZATION OF IDEALS IN ABELIAN EXTENSIONS OF NUMBER FIELDS

Abstract

We give a self-contained proof of a general conjecture of Gras on principalization of ideals in Abelian extensions of a given field L, which was solved by Kurihara in the case of totally real extensions L of the rational field ℚ. More precisely, for any given extension L/K of number fields, in which at least one infinite place of K totally splits, and for any ideal class cL of L, we construct a finite Abelian extension F/K, in which all infinite places totally split, such that cL become principal in the compositum M = LF.