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

Ichimura, Humio

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 779-786.

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 ${𝒪}_F^{\prime }\left[G\right]$ où ${𝒪}_F^{\prime }={𝒪}_F[1/p]$ 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 ${\zeta }_m\in F^{\times }$, nous donnons une condition nécessaire et suffisante pour que $N/F$ admette une $p$-NIB (Théorème 3). Lorsque ${\zeta }_m\notin F^{\times }$ et $p\nmid [F\left({\zeta }_m\right):F]$, 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 $[F\left({\zeta }_m\right):F]$, nous montrons que la propriété de descente n’est plus vraie en général (Théorème 2).

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, ${𝒪}_F^{\prime }={𝒪}_F[1/p]$ 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 $p\nmid [F\left({\zeta }_m\right):F]$, 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 $[F\left({\zeta }_m\right):F]$, we show that this descent property does not hold in general (Theorem 2).

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