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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,...

This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain $\sqrt{-1}$ (along with some partial results for the more general case). This catalog depends upon the , which as defined in [] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare and can occur only when the refined...

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....