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Search: MSC category 12J25 ( Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10] )

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1. CMB 2015 (vol 58 pp. 225)

Aghigh, Kamal; Nikseresht, Azadeh
Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials
Let $v$ be a henselian valuation of any rank of a field $K$ and $\overline{v}$ be the unique extension of $v$ to a fixed algebraic closure $\overline{K}$ of $K$. In 2005, it was studied properties of those pairs $(\theta,\alpha)$ of elements of $\overline{K}$ with $[K(\theta): K]\gt [K(\alpha): K]$ where $\alpha$ is an element of smallest degree over $K$ such that $$ \overline{v}(\theta-\alpha)=\sup\{\overline{v}(\theta-\beta) |\ \beta\in \overline{K}, \ [K(\beta): K]\lt [K(\theta): K]\}. $$ Such pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs.

Keywords:valued fields, non-Archimedean valued fields, irreducible polynomials
Categories:12J10, 12J25, 12E05

2. CMB 2011 (vol 55 pp. 233)

Bishnoi, Anuj; Khanduja, Sudesh K.
On Algebraically Maximal Valued Fields and Defectless Extensions
Let $v$ be a Henselian Krull valuation of a field $K$. In this paper, the authors give some necessary and sufficient conditions for a finite simple extension of $(K,v)$ to be defectless. Various characterizations of algebraically maximal valued fields are also given which lead to a new proof of a result proved by Yu. L. Ershov.

Keywords:valued fields, non-Archimedean valued fields
Categories:12J10, 12J25

3. CMB 2011 (vol 55 pp. 821)

Perez-Garcia, C.; Schikhof, W. H.
New Examples of Non-Archimedean Banach Spaces and Applications
The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm, without non-zero vectors that are $\|\cdot\|$-orthogonal and such that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$ into a valued field.

Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases
Categories:46S10, 12J25

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