Principalization of ideals by absolutely Abelian extensions

Let F/k be an abelian extension, with some conditions of signature for k and F, and let p be a prime number not dividing [F:k]. We prove, essentially, that there exists (infinitely many) p-extensions K of F, abelian over k, such that the p-class group Cl-F of F becomes trivial in K. These extensions K/F are effectively known and the structure of the group Gal(K/F) is directly connected with that of Cl-F. Numerical examples are given. (C) 1997 Academic Press.