Let $p$ be an odd prime number, $K$ a CM-field, and $K_{\infty}/K$
the cyclotomic $\mathbf{Z}_p$-extension with its $n$-th layer
$K_n$ ($n \geq 0$). Let $A_n$ be the Sylow $p$-subgroup of
the ideal class group of $K_n$. For the odd part $A_n^-$ of
$A_n$, it is well known that the natural map $A_n^- \to A_{n+1}^-$
is injective. The purpose of this note is to show that an analogous
phenomenon occurs for the Galois module structure of rings
of integers of a certain class of tamely ramified extensions
over $K_n$ of degree $p$.
Publié le : 2001-05-14
Classification:
Normal integral basis,
Kummer extension of prime degree,
CM-field,
$\mathbf {Z}_p$-extension,
11R33,
11R23
@article{1148393069,
author = {Ichimura, Humio},
title = {Note on the ring of integers of a Kummer extension of prime degree. III},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {77},
number = {10},
year = {2001},
pages = { 71-73},
language = {en},
url = {http://dml.mathdoc.fr/item/1148393069}
}
Ichimura, Humio. Note on the ring of integers of a Kummer extension of prime degree. III. Proc. Japan Acad. Ser. A Math. Sci., Tome 77 (2001) no. 10, pp. 71-73. http://gdmltest.u-ga.fr/item/1148393069/