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 C k,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, that is the 3 -class field tower of k , are drawn.