Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, pp. 183-204.

Nous étudions, en tant que module galoisien, le groupe des unités des extensions biquadratiques de corps de nombres L/M. Le 2-rang du premier groupe de cohomologie des unités de L/M est calculé pour M quelconque. Pour M quadratique imaginaire, nous déterminons la plupart des cas (incluant le cas L/M non ramifiée) où l’indice [V:V 1 V 2 V 3 ] prend sa valeur maximale 8, avec V les unités modulo la torsion de L et V i les unités modulo la torsion d’un des trois sous-corps quadratiques de L/M.

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.

DOI : 10.5802/jtnb.621
Mazur, Marcin 1 ; Ullom, Stephen V. 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
@article{JTNB_2008__20_1_183_0,
     author = {Mazur, Marcin and Ullom, Stephen V.},
     title = {Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {183--204},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {1},
     year = {2008},
     doi = {10.5802/jtnb.621},
     mrnumber = {2434163},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.621/}
}
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
SP  - 183
EP  - 204
VL  - 20
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.621/
DO  - 10.5802/jtnb.621
LA  - en
ID  - JTNB_2008__20_1_183_0
ER  - 
%0 Journal Article
%A Mazur, Marcin
%A Ullom, Stephen V.
%T Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields
%J Journal de théorie des nombres de Bordeaux
%D 2008
%P 183-204
%V 20
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.621/
%R 10.5802/jtnb.621
%G en
%F JTNB_2008__20_1_183_0
Mazur, Marcin; Ullom, Stephen V. Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, pp. 183-204. doi : 10.5802/jtnb.621. http://www.numdam.org/articles/10.5802/jtnb.621/

[1] G. Gras, Class Field Theory. Springer Monographs in Mathematics, Springer-Verlag, Berlin Heidelberg New York 2003. | MR | Zbl

[2] D. Harbater, Galois groups with prescribed ramification. Contemporary Math. 174 (1994), 35–60. | MR | Zbl

[3] H. Hasse, Über die Klassenzahl abelscher Zahlkörper. Springer-Verlag, Berlin Heidelberg New York Tokyo 1985. | Zbl

[4] M. Hirabayashi, K. Yoshino, Unit Indices of Imaginary Abelian Number Fields of Type (2,2,2). J. Number Th. 34 (1990), 346–361. | MR | Zbl

[5] F. Lemmermeyer, Kuroda’s class number formula. Acta Arith. 66 (1994), 245–260. | Zbl

[6] M. Mazur, S. V. Ullom, Galois module structure of units in real biquadratic number fields. Acta Arith. 111 (2004), 105–124. | MR | Zbl

Cité par Sources :