Maximal unramified extensions of imaginary quadratic number fields of small conductors

Ken Yamamura

Journal de théorie des nombres de Bordeaux (1997)

  • Volume: 9, Issue: 2, page 405-448
  • ISSN: 1246-7405

Abstract

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We determine the structures of the Galois groups Gal ( K u r / K ) of the maximal unramified extensions K u r of imaginary quadratic number fields K of conductors ≦ 420 ( ≦ 719 under the Generalized Riemann Hypothesis). For all such K , K u r is K , the Hilbert class field of K , the second Hilbert class field of K , or the third Hilbert class field of K . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class number relations and a computer for calculation of class numbers of fields of low degrees in order to get class numbers of fields of higher degrees. Results on class field towers and the knowledge of the 2 -groups of orders ≦ 2 6 and linear groups over finite fields are also used.

How to cite

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Yamamura, Ken. "Maximal unramified extensions of imaginary quadratic number fields of small conductors." Journal de théorie des nombres de Bordeaux 9.2 (1997): 405-448. <http://eudml.org/doc/248003>.

@article{Yamamura1997,
abstract = {We determine the structures of the Galois groups Gal$(K_\{ur\} / K)$ of the maximal unramified extensions $K_\{ur\}$ of imaginary quadratic number fields $K$ of conductors $\leqq 420 \, (\leqq 719$ under the Generalized Riemann Hypothesis). For all such $K$, $K_\{ur\}$ is $K$, the Hilbert class field of $K$, the second Hilbert class field of $K$, or the third Hilbert class field of $K$. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class number relations and a computer for calculation of class numbers of fields of low degrees in order to get class numbers of fields of higher degrees. Results on class field towers and the knowledge of the $2$-groups of orders $\leqq 2^6$ and linear groups over finite fields are also used.},
author = {Yamamura, Ken},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {maximal unramified extension; imaginary quadratic number field; discriminant bounds; class field tower; structure of Galois groups; maximal unramified extensions of imaginary quadratic number fields},
language = {eng},
number = {2},
pages = {405-448},
publisher = {Université Bordeaux I},
title = {Maximal unramified extensions of imaginary quadratic number fields of small conductors},
url = {http://eudml.org/doc/248003},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Yamamura, Ken
TI - Maximal unramified extensions of imaginary quadratic number fields of small conductors
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 2
SP - 405
EP - 448
AB - We determine the structures of the Galois groups Gal$(K_{ur} / K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 420 \, (\leqq 719$ under the Generalized Riemann Hypothesis). For all such $K$, $K_{ur}$ is $K$, the Hilbert class field of $K$, the second Hilbert class field of $K$, or the third Hilbert class field of $K$. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class number relations and a computer for calculation of class numbers of fields of low degrees in order to get class numbers of fields of higher degrees. Results on class field towers and the knowledge of the $2$-groups of orders $\leqq 2^6$ and linear groups over finite fields are also used.
LA - eng
KW - maximal unramified extension; imaginary quadratic number field; discriminant bounds; class field tower; structure of Galois groups; maximal unramified extensions of imaginary quadratic number fields
UR - http://eudml.org/doc/248003
ER -

References

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