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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 $𝔄$ 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 ${𝔓}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄(K\Gamma ;{𝔓}^h)$. 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 $𝔄(K\Gamma ;S)$ is a Hopf order and $S$ is $𝔄(K\Gamma ;S)$-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 ${𝔻}_{S/R}$ 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 $v_L\left(\rho \right)\equiv -v_L\left({𝔻}_{S/R}\right)-1(modn)$ 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 ${𝒜}_{L/K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_L$, the ring of integers of $L$, is then isomorphic to ${𝒜}_{L/K}$. This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that ${𝒜}_{L/K}$ 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 ${𝔽}_q\left[G\right]$ where ${𝔽}_q$ is the residue field) and its resulting ramification filtrations....

Given an algebraic number field $k$ and a finite group $\Gamma$, we write $ℛ\left(O_k\left[\Gamma \right]\right)$ 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 $ℛ\left(O_k\left[\Gamma \right]\right)$, 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 $\Gamma =V⋊C$, where $V$ is an elementary abelian group of order $p^r$ and $C$ is a cyclic group of order $m\>1$ acting faithfully on...