On the ring of $p$-integers of a cyclic $p$-extension over a number field

Ichimura, Humio. "On the ring of $p$-integers of a cyclic $p$-extension over a number field." Journal de Théorie des Nombres de Bordeaux 17.3 (2005): 779-786. <http://eudml.org/doc/249450>.

@article{Ichimura2005,
abstract = {Let $p$ be a prime number. A finite Galois extension $N/F$ of a number field $F$ with group $G$ has a normal $p$-integral basis ($p$-NIB for short) when $\{\mathcal\{O\}\}_N^\{\prime\}$ is free of rank one over the group ring $\{\mathcal\{O\}\}_F^\{\prime\}[G]$. Here, $\mathcal\{O\}_F^\{\prime\}=\mathcal\{O\}_F[1/p]$ is the ring of $p$-integers of $F$. Let $m=p^e$ be a power of $p$ and $N/F$ a cyclic extension of degree $m$. When $\zeta _m \in F^\{\times \}$, we give a necessary and sufficient condition for $N/F$ to have a $p$-NIB (Theorem 3). When $\zeta _m \notin F^\{\times \}$ and $p \nmid [F(\zeta _m) : F]$, we show that $N/F$ has a $p$-NIB if and only if $N(\zeta _m)/F(\zeta _m)$ has a $p$-NIB (Theorem 1). When $p$ divides $[F(\zeta _m) : F]$, we show that this descent property does not hold in general (Theorem 2).},
affiliation = {Faculty of Science Ibaraki University 2-1-1, Bunkyo, Mito, Ibaraki, 310-8512 Japan},
author = {Ichimura, Humio},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {779-786},
publisher = {Université Bordeaux 1},
title = {On the ring of $p$-integers of a cyclic $p$-extension over a number field},
url = {http://eudml.org/doc/249450},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Ichimura, Humio
TI - On the ring of $p$-integers of a cyclic $p$-extension over a number field
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 3
SP - 779
EP - 786
AB - Let $p$ be a prime number. A finite Galois extension $N/F$ of a number field $F$ with group $G$ has a normal $p$-integral basis ($p$-NIB for short) when ${\mathcal{O}}_N^{\prime}$ is free of rank one over the group ring ${\mathcal{O}}_F^{\prime}[G]$. Here, $\mathcal{O}_F^{\prime}=\mathcal{O}_F[1/p]$ is the ring of $p$-integers of $F$. Let $m=p^e$ be a power of $p$ and $N/F$ a cyclic extension of degree $m$. When $\zeta _m \in F^{\times }$, we give a necessary and sufficient condition for $N/F$ to have a $p$-NIB (Theorem 3). When $\zeta _m \notin F^{\times }$ and $p \nmid [F(\zeta _m) : F]$, we show that $N/F$ has a $p$-NIB if and only if $N(\zeta _m)/F(\zeta _m)$ has a $p$-NIB (Theorem 1). When $p$ divides $[F(\zeta _m) : F]$, we show that this descent property does not hold in general (Theorem 2).
LA - eng
UR - http://eudml.org/doc/249450
ER -