The class number one problem for some non-abelian normal CM-fields of degree $24$

Lemmermeyer, F.; Louboutin, S.; Okazaki, R.

The class number one problem for some non-abelian normal CM-fields of degree

24

Lemmermeyer, F. ; Louboutin, S. ; Okazaki, R.

Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 387-406.

Résumé
Abstract

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 $𝒜_{4} \times 𝒞_{2}$ (voir Théorème $14$ ), $(𝑖𝑖)$ qu’il y a au plus un tel corps de nombres de groupe de Galois $S L_{2} (𝔽_{3})$ (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 $𝒜_{4} \times 𝒞_{2}$ (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 $1$ .

MR Zbl

@article{JTNB_1999__11_2_387_0,
     author = {Lemmermeyer, F. and Louboutin, S. 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},
     mrnumber = {1745886},
     zbl = {1010.11063},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1999__11_2_387_0/}
}

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JO  - Journal de théorie des nombres de Bordeaux
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PB  - Université Bordeaux I
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%A Louboutin, S.
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%J Journal de théorie des nombres de Bordeaux
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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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