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**The class number one problem for the non-abelian normal CM-fields of degree 16.**
*(English)*
Zbl 0881.11079

In [S. Louboutin and R. Okazaki, The class number one problem for some non-abelian normal CM-fields of 2-power degrees, Proc. Lond. Math. Soc. (to appear)] the determination of all the non-abelian normal CM-fields of degree 16 with (relative) class number one was reduced to the determination of all the non-abelian normal CM-fields \(N=N_1N_2\) of degre 16 with (relative) class number one which are composita of two normal CM-fields \(N_1\) and \(N_2\) with the same maximal real subfield. All such composita were determined which have relative class number one for which at least one of the \(N_i\)’s is a quaternion octic CM-field.

Here, the author determines octic CM-fields. There are two cases to be considered. First, \(N_1\) and \(N_2\) are both dihedral octic CM-fields, in which case he proves that there are two such composita which have relative class number one, and both of them have class number one. Second, one of the \(N_i\)’s is a dihedral octic CM-field and the other is an imaginary abelian octic field, in which case he proves that there are four such composite which have class number one. It is worth mentioning that this first determination stems from good lower bounds on relative class numbers of dihedral octic CM-fields and on non-normal quartic CM-fields.

Here, the author determines octic CM-fields. There are two cases to be considered. First, \(N_1\) and \(N_2\) are both dihedral octic CM-fields, in which case he proves that there are two such composita which have relative class number one, and both of them have class number one. Second, one of the \(N_i\)’s is a dihedral octic CM-field and the other is an imaginary abelian octic field, in which case he proves that there are four such composite which have class number one. It is worth mentioning that this first determination stems from good lower bounds on relative class numbers of dihedral octic CM-fields and on non-normal quartic CM-fields.

Reviewer: S.Louboutin (Caen)

### MSC:

11R29 | Class numbers, class groups, discriminants |

11R21 | Other number fields |

11R42 | Zeta functions and \(L\)-functions of number fields |

11M20 | Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\) |