The generators of $3$-class group of some fields of degree $6$ over $\mathbb{Q}$

Abstract

Let be k=Q(\sqrt[3]{p},\zeta_3),
 where p is a prime number such that p \equiv 1 (mod 9), 
 and let C_{k,3} the 3-component of the class group of k. 
 In his work [7], Frank Gerth III proves a conjecture made by Calegari and Emerton which gives a necessary and sufficient conditions for C_{k,3} to be of rank two.
 The present work display a consideration steps towards determination of generators of C_{k,3}, when C_{k,3} is isomorphic to Z/9Z \times Z/3Z.

$3$-Principalization over $S_3$-fields

“…7. )(2) The absolute Galois group Gal(k/Q) ≃ S 3 of k acts on the 3-class group C k,3 ≃ G/G ′ and thus also on the Frattini quotient Q = G/Φ(G) = G/(G ′ • G 3 ), whence Aut(Q) contains a subgroup isomorphic to S 3 = 6, 1 .…”