3-Principalization over S3 -fields

Abstract

Let p ≡ 1 (mod 9) be a prime number and ζ3 be a primitive cube root of unity. Then k = Q( √3 p, ζ3) is a pure metacyclic field with group Gal(k/Q) ≃ S3 . In the case that k possesses a 3-class group Ck,3 of type (9, 3), the capitulation of 3-ideal classes of k in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-3-extension k (∞) 3 , that is the 3-class field tower of k , are drawn