The class number one problem for some non-abelian normal CM-fields of degree 24
Journal de Théorie des Nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 387-406.

Nous déterminons tous les corps de nombres de degré 24, galoisiens mais non-abéliens, à multiplication complexe et tels que les groupes de Galois de leurs sous-corps totalement réels maximaux soient isomorphes à 𝒜 4 (le groupe alterné de degré 4 et d’ordre 12) qui sont de nombres de classes d’idéaux égaux à 1. Nous prouvons (𝑖) qu’il y a deux tels corps de nombres de groupes de Galois 𝒜 4 ×𝒞 2 (voir Théorème 14), (𝑖𝑖) qu’il y a au plus un tel corps de nombres de groupe de Galois SL 2 (𝔽 3 ) (voir Théorème 18), et (𝑖𝑖𝑖) que sous l’hypothèse de Riemann généralisée ce dernier corps candidat est effectivement de nombre de classes d’idéaux égal à 1.

We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to 𝒜 4 , the alternating group of degree 4 and order 12. There are two such fields with Galois group 𝒜 4 ×𝒞 2 (see Theorem 14) and at most one with Galois group SL 2 (𝔽 3 ) (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number 1.

@article{JTNB_1999__11_2_387_0,
     author = {Lemmermeyer, Franz and Louboutin, St\'ephane R. and Okazaki, R.},
     title = {The class number one problem for some non-abelian normal CM-fields of degree $24$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {387--406},
     publisher = {Universit\'e Bordeaux I},
     volume = {11},
     number = {2},
     year = {1999},
     zbl = {1010.11063},
     mrnumber = {1745886},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1999__11_2_387_0/}
}
Lemmermeyer, F.; Louboutin, S.; Okazaki, R. The class number one problem for some non-abelian normal CM-fields of degree $24$. Journal de Théorie des Nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 387-406. http://www.numdam.org/item/JTNB_1999__11_2_387_0/

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