On the ring of $p$-integers of a cyclic $p$-extension over a number field
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 779-786.

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 ${𝒪}_{N}^{\prime }$ is free of rank one over the group ring ${𝒪}_{F}^{\prime }\left[G\right]$. Here, ${𝒪}_{F}^{\prime }={𝒪}_{F}\left[1/p\right]$ 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 ${\zeta }_{m}\in {F}^{×}$, we give a necessary and sufficient condition for $N/F$ to have a $p$-NIB (Theorem 3). When ${\zeta }_{m}\notin {F}^{×}$ and $p\nmid \left[F\left({\zeta }_{m}\right):F\right]$, we show that $N/F$ has a $p$-NIB if and only if $N\left({\zeta }_{m}\right)/F\left({\zeta }_{m}\right)$ has a $p$-NIB (Theorem 1). When $p$ divides $\left[F\left({\zeta }_{m}\right):F\right]$, we show that this descent property does not hold in general (Theorem 2).

Soit $p$ un nombre premier. On dit qu’une extension finie, galoisienne, $N/F$ d’un corps de nombres $F$, à groupe de Galois $G$, admet une base normale $p$-entière ($p$-NIB en abrégé) si ${𝒪}_{N}^{\prime }$ est libre de rang un sur l’anneau de groupe ${𝒪}_{F}^{\prime }\left[G\right]$${𝒪}_{F}^{\prime }={𝒪}_{F}\left[1/p\right]$ désigne l’anneau des $p$-entiers de $F$. Soit $m={p}^{e}$ une puissance de $p$ et $N/F$ une extension cyclique de degré $m$. Lorsque ${\zeta }_{m}\in {F}^{×}$, nous donnons une condition nécessaire et suffisante pour que $N/F$ admette une $p$-NIB (Théorème 3). Lorsque ${\zeta }_{m}\notin {F}^{×}$ et $p\nmid \left[F\left({\zeta }_{m}\right):F\right]$, nous montrons que $N/F$ admet une $p$-NIB si et seulement si $N\left({\zeta }_{m}\right)/F\left({\zeta }_{m}\right)$ admet $p$-NIB (Théorème 1). Enfin, si $p$ divise $\left[F\left({\zeta }_{m}\right):F\right]$, nous montrons que la propriété de descente n’est plus vraie en général (Théorème 2).

DOI: 10.5802/jtnb.520
Humio Ichimura 1

1 Faculty of Science Ibaraki University 2-1-1, Bunkyo, Mito, Ibaraki, 310-8512 Japan
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Humio Ichimura. On the ring of $p$-integers of a cyclic $p$-extension over a number field. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 779-786. doi : 10.5802/jtnb.520. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.520/

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