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.