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

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.

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.

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F. Lemmermeyer; S. Louboutin; R. Okazaki. The class number one problem for some non-abelian normal CM-fields of degree $24$. Journal de Théorie des Nombres de Bordeaux, Volume 11 (1999) no. 2, pp. 387-406. doi : 10.5802/jtnb.257. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.257/

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