Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields

Marcin Mazur; Stephen V. Ullom

doi:10.5802/jtnb.621

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, Tome 20 (2008) no. 1, pp. 183-204.

Résumé
Abstract

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 $L / M$ .

MR

DOI : 10.5802/jtnb.621

Affiliations des auteurs :

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

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Marcin Mazur; Stephen V. Ullom. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.621/

Bibliographie
Cité par

[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 :