Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields

Marcin Mazur[1]; Stephen V. Ullom[2]

  • [1] Department of Mathematics Binghamton University P.O. Box 6000 Binghamton, NY 13892-6000
  • [2] Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, Illinois 61801-2975

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 1, page 183-204
  • ISSN: 1246-7405

Abstract

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We investigate as Galois module the unit group of biquadratic extensions L / M of number fields. The 2 -rank of the first cohomology group of units of L / M is computed for general M . For M imaginary quadratic we determine a large portion of the cases (including all unramified L / M ) where the index [ V : V 1 V 2 V 3 ] takes its maximum value 8 , where V are units mod torsion of L and V i are units mod torsion of one of the 3 quadratic subfields of L / M .

How to cite

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Mazur, Marcin, and Ullom, Stephen V.. "Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 183-204. <http://eudml.org/doc/10827>.

@article{Mazur2008,
abstract = {We investigate as Galois module the unit group of biquadratic extensions $L/M$ of number fields. The $2$-rank of the first cohomology group of units of $L/M$ is computed for general $M$. For $M$ imaginary quadratic we determine a large portion of the cases (including all unramified $L/M$) where the index $[V:V_1 V_2 V_3]$ takes its maximum value $8$, where $V$ are units mod torsion of $L$ and $V_i$ are units mod torsion of one of the 3 quadratic subfields of $L/M$.},
affiliation = {Department of Mathematics Binghamton University P.O. Box 6000 Binghamton, NY 13892-6000; Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, Illinois 61801-2975},
author = {Mazur, Marcin, Ullom, Stephen V.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {183-204},
publisher = {Université Bordeaux 1},
title = {Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields},
url = {http://eudml.org/doc/10827},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Mazur, Marcin
AU - Ullom, Stephen V.
TI - Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 1
SP - 183
EP - 204
AB - We investigate as Galois module the unit group of biquadratic extensions $L/M$ of number fields. The $2$-rank of the first cohomology group of units of $L/M$ is computed for general $M$. For $M$ imaginary quadratic we determine a large portion of the cases (including all unramified $L/M$) where the index $[V:V_1 V_2 V_3]$ takes its maximum value $8$, where $V$ are units mod torsion of $L$ and $V_i$ are units mod torsion of one of the 3 quadratic subfields of $L/M$.
LA - eng
UR - http://eudml.org/doc/10827
ER -

References

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  1. G. Gras, Class Field Theory. Springer Monographs in Mathematics, Springer-Verlag, Berlin Heidelberg New York 2003. Zbl1019.11032MR1941965
  2. D. Harbater, Galois groups with prescribed ramification. Contemporary Math. 174 (1994), 35–60. Zbl0815.11053MR1299733
  3. H. Hasse, Über die Klassenzahl abelscher Zahlkörper. Springer-Verlag, Berlin Heidelberg New York Tokyo 1985. Zbl0668.12004
  4. M. Hirabayashi, K. Yoshino, Unit Indices of Imaginary Abelian Number Fields of Type ( 2 , 2 , 2 ) . J. Number Th. 34 (1990), 346–361. Zbl0705.11065MR1049510
  5. F. Lemmermeyer, Kuroda’s class number formula. Acta Arith. 66 (1994), 245–260. Zbl0807.11052
  6. M. Mazur, S. V. Ullom, Galois module structure of units in real biquadratic number fields. Acta Arith. 111 (2004), 105–124. Zbl1060.11070MR2039416

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