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

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⩾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⩾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.