Formal groups and the Galois module structure of local rings of integers. (English) Zbl 0582.12008

The goal of this article is to describe as Galois modules, the ring of integers of certain local extensions. Let \({\mathbb{Q}}_ p\) denote the p- adic rational field, K is a finite extension of \({\mathbb{Q}}_ p\), \(\lambda\) a uniformizing parameter of K and \(q=p^ n\) the absolute norm of \({\mathfrak p}_ K\). For certain totally ramified abelian extensions \(N=K(\pi)\) and \(L=K(\pi^{[\lambda^ m]})\) of degree \(q^{m+r-1}(q-1)\) and \(q^ r(q- 1)\) over K, let \(\Gamma =Gal(N/L)\). Here \(\pi^{[\lambda^ m]}\) denotes a certain power series with \(\pi\) as the variable.
The group ring \(L\Gamma\) induces the obvious action on N. Let \({\mathfrak O}_ N\) denote the ring of integers of N and \({\mathfrak A}=\{x\in L\Gamma |\) \({\mathfrak O}_ Nx\subseteq {\mathfrak O}_ N\}\). Then the author proves that \({\mathfrak O}_ N\) is \({\mathfrak A}\)-free on any element of \(N^{\times}\) with \({\mathfrak p}_ N\) valuation \(q^ m-1\). In particular, \({\mathfrak O}_ N=\pi^{[\lambda^ m]-1}{\mathfrak A}.\)
These results resemble those obtained by L. R. McCulloh [J. Algebra 82, 102-134 (1983; Zbl 0508.12008)].
Reviewer: Ch.Parry


11S20 Galois theory
11S45 Algebras and orders, and their zeta functions
14L05 Formal groups, \(p\)-divisible groups


Zbl 0508.12008
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