For the cyclotomic -extension of an imaginary quadratic field , we consider the Galois group of the maximal unramified pro--extension over . In this paper, we give some families of for which is a metabelian pro--group with the explicit presentation, and determine the case that becomes a nonabelian metacyclic pro--group. We also calculate Iwasawa theoretically the Galois groups of -class field towers of certain cyclotomic -extensions.
Pour quadratique imaginaire, nous étudions le groupe de Galois de la pro--extension non ramifiée maximale au-dessus de la -extension cyclotomique de . Nous déterminons des familles de tels corps imaginaires pour lesquels est un pro--groupe métabélien et en donnons une présentation explicite ; nous précisons de même des familles pour lesquelles est un pro--groupe métacyclique non abélien. Nous calculons enfin en termes de Théorie d’Iwasawa les groupes de Galois de -tours de corps de classes de certaines -extensions cyclotomiques.
@article{JTNB_2010__22_1_115_0, author = {Mizusawa, Yasushi}, title = {On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {115--138}, publisher = {Universit\'e Bordeaux 1}, volume = {22}, number = {1}, year = {2010}, doi = {10.5802/jtnb.707}, zbl = {1221.11215}, mrnumber = {2675876}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.707/} }
TY - JOUR AU - Mizusawa, Yasushi TI - On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 115 EP - 138 VL - 22 IS - 1 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.707/ DO - 10.5802/jtnb.707 LA - en ID - JTNB_2010__22_1_115_0 ER -
%0 Journal Article %A Mizusawa, Yasushi %T On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field %J Journal de théorie des nombres de Bordeaux %D 2010 %P 115-138 %V 22 %N 1 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.707/ %R 10.5802/jtnb.707 %G en %F JTNB_2010__22_1_115_0
Mizusawa, Yasushi. On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 115-138. doi : 10.5802/jtnb.707. http://archive.numdam.org/articles/10.5802/jtnb.707/
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