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 , the alternating group of degree and order . There are two such fields with Galois group (see Theorem 14) and at most one with Galois group SL (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number .
Nous déterminons tous les corps de nombres de degré , galoisiens mais non-abéliens, à multiplication complexe et tels que les groupes de Galois de leurs sous-corps totalement réels maximaux soient isomorphes à (le groupe alterné de degré et d’ordre ) qui sont de nombres de classes d’idéaux égaux à . Nous prouvons qu’il y a deux tels corps de nombres de groupes de Galois (voir Théorème ), qu’il y a au plus un tel corps de nombres de groupe de Galois (voir Théorème 18), et que sous l’hypothèse de Riemann généralisée ce dernier corps candidat est effectivement de nombre de classes d’idéaux égal à .
@article{JTNB_1999__11_2_387_0, author = {F. Lemmermeyer and S. Louboutin and R. Okazaki}, 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 = {https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_387_0/} }
TY - JOUR AU - F. Lemmermeyer AU - S. Louboutin AU - R. Okazaki TI - The class number one problem for some non-abelian normal CM-fields of degree $24$ JO - Journal de théorie des nombres de Bordeaux PY - 1999 SP - 387 EP - 406 VL - 11 IS - 2 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_387_0/ LA - en ID - JTNB_1999__11_2_387_0 ER -
%0 Journal Article %A F. Lemmermeyer %A S. Louboutin %A R. Okazaki %T The class number one problem for some non-abelian normal CM-fields of degree $24$ %J Journal de théorie des nombres de Bordeaux %D 1999 %P 387-406 %V 11 %N 2 %I Université Bordeaux I %U https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_387_0/ %G en %F JTNB_1999__11_2_387_0
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. https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_387_0/
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