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Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials

  Published:2015-02-10
 Printed: Jun 2015
  • Kamal Aghigh,
    Department of Mathematics, K. N. Toosi University of Technology, P.O.Box 16315-1618, Tehran, Iran
  • Azadeh Nikseresht,
    Department of Mathematics, K. N. Toosi University of Technology, P.O.Box 16315-1618, Tehran, Iran
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Abstract

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 valued fields, non-Archimedean valued fields, irreducible polynomials
MSC Classifications: 12J10, 12J25, 12E05 show english descriptions Valued fields
Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10]
Polynomials (irreducibility, etc.)
12J10 - Valued fields
12J25 - Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10]
12E05 - Polynomials (irreducibility, etc.)
 

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