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**Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one.**
*(English)*
Zbl 0809.11069

In a previous paper [Tôhoku Math. J., II. Ser. 46, No. 1, 1-12 (1994; Zbl 0796.11050)] the first author proved that a non normal quartic number field \(K\) is a CM-field with relative class number one if and only if its normal closure \(N\) is a dihedral octic CM-field with relative class number one. Moreover, he proved that if \(K\) has relative class number one then \(d_ K/d_ k\leq 5\cdot 10^ 9\), where \(d_ K\) is the discriminant of \(K\) and \(d_ k\) is the discriminant of its real quadratic subfield \(k\).

In this paper the authors make use of this upper bound, of a characterization of the non-normal quartic CM-fields with odd relative class number, and of a necessary condition for the relative class number of any CM-field to be one, to get a short list of number fields that contains all non-normal quartic CM-fields with relative class number one. Then, they compute the relative class numbers of the number fields of this list, the class numbers of their maximal real subfields and the class numbers of the maximal real subfields of their normal closures. Finally, they notice that quaternion octic CM-fields have even relative class numbers. Hence, they settle the desired determination and get that there are 37 non-isomorphic non-normal quartic CM-fields with class number one, and 17 non-abelian normal octic CM-fields with class number one.

In this paper the authors make use of this upper bound, of a characterization of the non-normal quartic CM-fields with odd relative class number, and of a necessary condition for the relative class number of any CM-field to be one, to get a short list of number fields that contains all non-normal quartic CM-fields with relative class number one. Then, they compute the relative class numbers of the number fields of this list, the class numbers of their maximal real subfields and the class numbers of the maximal real subfields of their normal closures. Finally, they notice that quaternion octic CM-fields have even relative class numbers. Hence, they settle the desired determination and get that there are 37 non-isomorphic non-normal quartic CM-fields with class number one, and 17 non-abelian normal octic CM-fields with class number one.

Reviewer: S.Louboutin (Caen)

### MSC:

11R29 | Class numbers, class groups, discriminants |

11R21 | Other number fields |

11R16 | Cubic and quartic extensions |