On Galois structure of the integers in cyclic extensions of local number fields

Elder, G. Griffith

Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 113-149

Abstract (VO)

Let $p$ be a rational prime, $K$ be a finite extension of the field of $p$-adic numbers, and let $L/K$ be a totally ramified cyclic extension of degree $p^n$. Restrict the first ramification number of $L/K$ to about half of its possible values, $b_1>1/2·p{e}_0/(p-1)$ where $e_0$ denotes the absolute ramification index of $K$. Under this loose condition, we explicitly determine the ${\mathbb{Z}}_p\left[G\right]$-module structure of the ring of integers of $L$, where ${\mathbb{Z}}_p$ denotes the $p$-adic integers and $G$ denotes the Galois group Gal$(L/K)$. In the process of determining this structure, we study various restrictions on the ramification filtration and examine the trace map relationships that result. Two of these restrictions are generalizations of almost maximal ramification. Our method for determining this structure is constructive (also inductive). We exhibit generators for the ring of integers of $L$ over the group ring, ${\mathbb{Z}}_p\left[G\right]$ (actually over ${𝔒}_T\left[G\right]$ where ${𝔒}_T$ is the ring of integers in the maximal unramified subfield of $K$). They are determined in an essential way by their valuation. Then we describe their relations.

Résumé

Soit $p$ un nombre premier, $K$ une extension finie du corps des nombres $p$-adiques, et $L/K$ une extension cyclique ramifiée de degré $p^n$. On suppose que le premier nombre de ramification de $L/K$ satisfait $b_1>1/2·p{e}_0/(p-1)$ où $e_0$ est l’indice de ramification absolu de $K$. Nous déterminons explicitement la structure de l’anneau des entiers de $L$ comme ${\mathbb{Z}}_p\left[G\right]$-module, où ${\mathbb{Z}}_p$ désigne l’anneau des entiers $p$-adiques et $G=\text{Gal}(L/K)$ le groupe de Galois de $L$.