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**The class number one problem for some non-abelian normal CM-fields of 2-power degrees.**
*(English)*
Zbl 0891.11054

The authors begin by proving that a non-abelian normal CM-field of degree 16 has odd relative class number if and only if it is dihedral, or is a composition of two normal octic CM-fields with the same maximal totally real subfield, or has Galois group
\[
G_9=\langle a,b,z;\;a^2=b^2=z^4=1,\;b^{- 1}ab=az^2,\;az=za,\;bz=zb\rangle.
\]
They then solve several relative class number one problems:

1) They solve the relative class number one problem for the dihedral CM-fields of 2-power degrees. First, a reminder of the characterization of the dihedral CM-fields of 2-power degrees with odd relative class numbers is given. Second, lower bounds on relative class numbers of dihedral CM-fields of 2-power degrees with odd relative class numbers are given, thus obtaining an upper bound on the discriminants of the dihedral CM-fields of 2-power degrees with relative class number equal to one. Third, they compute the relative class numbers of all the dihedral CM-fields of 2-power degrees with odd relative class numbers and discriminants less than or equal to this latter bound. They end up with a list of 24 dihedral CM-fields of 2-power degrees with relative class number one and show that exactly 21 out of them have class number one.

2) They determine all the non-abelian normal CM-fields of degree 16 with Galois group \(G_9\) which have relative class number one (there is only one such number field), and then which have class number one (there is only one such number field).

3) They determine some of the non-abelian normal CM-fields \(N=N_1N_2\) of degree 16 which are composita of two normal octic CM-fields \(N_1\) and \(N_2\) with the same maximal totally real subfield which have relative class number one, and then which have class number one. Indeed, they focus on the case where one of the \(N_i\)’s is a quaternion octic CM-field and using [S. Louboutin, J. Lond. Math. Soc., II. Ser. 54, 227-238 (1996; Zbl 0861.11064)], they prove that there is only one such composita with relative class number one and this compositum has class number one.

According to this paper, to [S. Louboutin, R. Okazaki and M. Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. Am. Math. Soc. 349, 3657–3678 (1997; Zbl 0893.11045)] and to [S. Louboutin, The class number one problem for the non-abelian normal CM-fields of degree 16, Acta Arith. 82, 173–196 (1997; Zbl 0881.11079)] the determination of all the non-abelian normal CM-fields of degree less than 20 which have class number one is complete.

1) They solve the relative class number one problem for the dihedral CM-fields of 2-power degrees. First, a reminder of the characterization of the dihedral CM-fields of 2-power degrees with odd relative class numbers is given. Second, lower bounds on relative class numbers of dihedral CM-fields of 2-power degrees with odd relative class numbers are given, thus obtaining an upper bound on the discriminants of the dihedral CM-fields of 2-power degrees with relative class number equal to one. Third, they compute the relative class numbers of all the dihedral CM-fields of 2-power degrees with odd relative class numbers and discriminants less than or equal to this latter bound. They end up with a list of 24 dihedral CM-fields of 2-power degrees with relative class number one and show that exactly 21 out of them have class number one.

2) They determine all the non-abelian normal CM-fields of degree 16 with Galois group \(G_9\) which have relative class number one (there is only one such number field), and then which have class number one (there is only one such number field).

3) They determine some of the non-abelian normal CM-fields \(N=N_1N_2\) of degree 16 which are composita of two normal octic CM-fields \(N_1\) and \(N_2\) with the same maximal totally real subfield which have relative class number one, and then which have class number one. Indeed, they focus on the case where one of the \(N_i\)’s is a quaternion octic CM-field and using [S. Louboutin, J. Lond. Math. Soc., II. Ser. 54, 227-238 (1996; Zbl 0861.11064)], they prove that there is only one such composita with relative class number one and this compositum has class number one.

According to this paper, to [S. Louboutin, R. Okazaki and M. Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. Am. Math. Soc. 349, 3657–3678 (1997; Zbl 0893.11045)] and to [S. Louboutin, The class number one problem for the non-abelian normal CM-fields of degree 16, Acta Arith. 82, 173–196 (1997; Zbl 0881.11079)] the determination of all the non-abelian normal CM-fields of degree less than 20 which have class number one is complete.

Reviewer: S. Louboutin (Caen)