Unit indices and cohomology for biquadratic extensions of imaginary quadratic fields
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, p. 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.
@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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {1},
     year = {2008},
     pages = {183-204},
     doi = {10.5802/jtnb.621},
     mrnumber = {2434163},
     zbl = {pre05543196},
     language = {en},
     url = {http://www.numdam.org/item/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. https://www.numdam.org/item/JTNB_2008__20_1_183_0/

[1] G. Gras, Class Field Theory. Springer Monographs in Mathematics, Springer-Verlag, Berlin Heidelberg New York 2003. | MR 1941965 | Zbl 1019.11032

[2] D. Harbater, Galois groups with prescribed ramification. Contemporary Math. 174 (1994), 35–60. | MR 1299733 | Zbl 0815.11053

[3] H. Hasse, Über die Klassenzahl abelscher Zahlkörper. Springer-Verlag, Berlin Heidelberg New York Tokyo 1985. | Zbl 0668.12004

[4] M. Hirabayashi, K. Yoshino, Unit Indices of Imaginary Abelian Number Fields of Type (2,2,2). J. Number Th. 34 (1990), 346–361. | MR 1049510 | Zbl 0705.11065

[5] F. Lemmermeyer, Kuroda’s class number formula. Acta Arith. 66 (1994), 245–260. | Zbl 0807.11052

[6] M. Mazur, S. V. Ullom, Galois module structure of units in real biquadratic number fields. Acta Arith. 111 (2004), 105–124. | MR 2039416 | Zbl 1060.11070