# 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

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topMazur, 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

top- G. Gras, Class Field Theory. Springer Monographs in Mathematics, Springer-Verlag, Berlin Heidelberg New York 2003. Zbl1019.11032MR1941965
- D. Harbater, Galois groups with prescribed ramification. Contemporary Math. 174 (1994), 35–60. Zbl0815.11053MR1299733
- H. Hasse, Über die Klassenzahl abelscher Zahlkörper. Springer-Verlag, Berlin Heidelberg New York Tokyo 1985. Zbl0668.12004
- 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
- F. Lemmermeyer, Kuroda’s class number formula. Acta Arith. 66 (1994), 245–260. Zbl0807.11052
- 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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