Let $k$ be a finite extension of ${\mathbb{Q}}_{p}$, let ${k}_{1}$, respectively ${k}_{3}$, be the division fields of level $1$, respectively $3$, arising from a Lubin-Tate formal group over $k$, and let $\Gamma =$ Gal(${k}_{3}/{k}_{1}$). It is known that the valuation ring ${k}_{3}$ cannot be free over its associated order $\U0001d504$ in $K\Gamma $ unless $k={\mathbb{Q}}_{p}$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.

Let $K$ be a finite extension of ${\mathbb{Q}}_{p}$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma $ and ramification index ${p}^{n}$. We give a criterion in terms of the ramification numbers ${t}_{i}$ for a fractional ideal ${\U0001d513}^{h}$ of the valuation ring $S$ of $L$ not to be free over its associated order $\U0001d504(K\Gamma ;{\U0001d513}^{h})$. In particular, if ${t}_{n}-[{t}_{n}/p]\<{p}^{n-1}e$ then the inverse different can be free over its associated order only when ${t}_{i}\equiv -1$ (mod ${p}^{n}$) for all $i$. We give three consequences of this. Firstly, if $\U0001d504(K\Gamma ;S)$ is a Hopf order and $S$ is $\U0001d504(K\Gamma ;S)$-Galois...

Let $S/R$ be a finite extension of discrete valuation rings of characteristic $p\>0$, and suppose that the corresponding extension $L/K$ of fields of fractions is separable and is $H$-Galois for some $K$-Hopf algebra $H$. Let ${\mathbb{D}}_{S/R}$ be the different of $S/R$. We show that if $S/R$ is totally ramified and its degree $n$ is a power of $p$, then any element $\rho $ of $L$ with ${v}_{L}\left(\rho \right)\equiv -{v}_{L}\left({\mathbb{D}}_{S/R}\right)-1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n)$ generates $L$ as an $H$-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

Let $L/K$ be an extension of algebraic number fields, where $L$ is abelian over $\mathbb{Q}$. In this paper we give an explicit description of the associated order ${\mathcal{A}}_{L/K}$ of this extension when $K$ is a cyclotomic field, and prove that ${o}_{L}$, the ring of integers of $L$, is then isomorphic to ${\mathcal{A}}_{L/K}$. This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that ${\mathcal{A}}_{L/K}$ is the maximal order if $L/K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to ${\mathbb{Q}}^{\left({m}^{\text{'}}\right)}/K$, where ${m}^{\text{'}}$ is the conductor of $K$.

Which invariants of a Galois $p$-extension of local number fields $L/K$ (residue field of char $p$, and Galois group $G$) determine the structure of the ideals in $L$ as modules over the group ring ${\mathbb{Z}}_{p}\left[G\right]$, ${\mathbb{Z}}_{p}$ the $p$-adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups $G$, we propose and study a new group (within the group ring ${\mathbb{F}}_{q}\left[G\right]$ where ${\mathbb{F}}_{q}$ is the residue field) and its resulting ramification filtrations....

Given an algebraic number field $k$ and a finite group $\Gamma $, we write $\mathcal{R}\left({O}_{k}\left[\Gamma \right]\right)$ for the subset of the locally free classgroup $\mathrm{Cl}\left({O}_{k}\left[\Gamma \right]\right)$ consisting of the classes of rings of integers ${O}_{N}$ in tame Galois extensions $N/k$ with $\mathrm{Gal}(N/k)\cong \Gamma $. We determine $\mathcal{R}\left({O}_{k}\left[\Gamma \right]\right)$, and show it is a subgroup of $\mathrm{Cl}\left({O}_{k}\left[\Gamma \right]\right)$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $\Gamma =V\u22caC$, where $V$ is an elementary abelian group of order ${p}^{r}$ and $C$ is a cyclic group of order $m\>1$ acting faithfully on...

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