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

F. Lemmermeyer; S. Louboutin; R. Okazaki

Journal de théorie des nombres de Bordeaux (1999)

  • Volume: 11, Issue: 2, page 387-406
  • ISSN: 1246-7405

Abstract

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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 .

How to cite

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Lemmermeyer, F., Louboutin, S., and Okazaki, R.. "The class number one problem for some non-abelian normal CM-fields of degree $24$." Journal de théorie des nombres de Bordeaux 11.2 (1999): 387-406. <http://eudml.org/doc/248333>.

@article{Lemmermeyer1999,
abstract = {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 $\mathcal \{A\}_4$, the alternating group of degree $4$ and order $12$. There are two such fields with Galois group $\mathcal \{A\}_4 \times \mathcal \{C\}_2$ (see Theorem 14) and at most one with Galois group SL$_2(\mathbb \{F\}_3)$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$.},
author = {Lemmermeyer, F., Louboutin, S., Okazaki, R.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {nonabelian normal CM-fields; class number one; Galois group},
language = {eng},
number = {2},
pages = {387-406},
publisher = {Université Bordeaux I},
title = {The class number one problem for some non-abelian normal CM-fields of degree $24$},
url = {http://eudml.org/doc/248333},
volume = {11},
year = {1999},
}

TY - JOUR
AU - Lemmermeyer, F.
AU - Louboutin, S.
AU - Okazaki, R.
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
PB - Université Bordeaux I
VL - 11
IS - 2
SP - 387
EP - 406
AB - 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 $\mathcal {A}_4$, the alternating group of degree $4$ and order $12$. There are two such fields with Galois group $\mathcal {A}_4 \times \mathcal {C}_2$ (see Theorem 14) and at most one with Galois group SL$_2(\mathbb {F}_3)$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$.
LA - eng
KW - nonabelian normal CM-fields; class number one; Galois group
UR - http://eudml.org/doc/248333
ER -

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