We study, for any prime number $p$, the triviality of certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rational field. Among others, we prove that if $p$ is $2$ or $3$ and $l$ is a prime number not congruent to $1$ or $-1$ modulo $2p^2$, then $l$ does not divide the class number of the cyclotomic field of $p^u$th roots of unity for any positive integer $u$.
Publié le : 2007-05-14
Classification:
Ideal class group,
$\boldsymbol{Z}_p$-extension,
cyclotomic field,
class number formula,
decomposition field,
11R29,
11R18,
11R20,
11R23
@article{1182180736,
author = {Horie, Kuniaki},
title = {Certain primary components of the ideal class group of the $\boldsymbol{Z}\_p$-extension over the rationals},
journal = {Tohoku Math. J. (2)},
volume = {59},
number = {1},
year = {2007},
pages = { 259-291},
language = {en},
url = {http://dml.mathdoc.fr/item/1182180736}
}
Horie, Kuniaki. Certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rationals. Tohoku Math. J. (2), Tome 59 (2007) no. 1, pp. 259-291. http://gdmltest.u-ga.fr/item/1182180736/