The class number one problem for some non-normal CM-fields of degree $2p$

Jeoung-Hwan Ahn (1) , Gérard Boutteaux (2) , Soun-Hi Kwon (1) , Stéphane Louboutin (3)

1 Korea University
2 UP13 - Université Paris 13
3 IML - Institut de mathématiques de Luminy

To date, the class number one problem for non-normal CM-fields is solved only for quartic CM-fields. Here, we solve it for a family of non-normal CM-fields of degree 2p , p⩾3p⩾3 and odd prime. We determine all the non-isomorphic non-normal CM-fields of degree 2p, containing a real cyclic field of degree p , and of class number one. Here, p⩾3p⩾3 ranges over the odd primes. There are 24 such non-isomorphic number fields: 19 of them are of degree 6 and 5 of them are of degree 10. We also construct 19 non-isomorphic non-normal CM-fields of degree 12 and of class number one, and 10 non-isomorphic non-normal CM-fields of degree 20 and of class number one.

Mots clés: CM-field; class number; Dedekind zeta function

Domaines: Théorie des nombres [math.NT]

Jeoung-Hwan Ahn, Gérard Boutteaux, Soun-Hi Kwon, Stéphane Louboutin. The class number one problem for some non-normal CM-fields of degree $2p$. Journal of Number Theory, 2012, 132 (8), pp.1793-1806. ⟨10.1016/j.jnt.2012.02.020⟩. ⟨hal-01286358⟩