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Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials

Published online by Cambridge University Press:  20 November 2018

Kamal Aghigh  and
Azadeh Nikseresht
Kamal Aghigh
Affiliation:
Department of Mathematics, K. N. Toosi University of Technology, P.O.Box 16315-1618, Tehran, Iran. e-mail: [email protected] e-mail: [email protected]
Azadeh Nikseresht
Affiliation:
Department of Mathematics, K. N. Toosi University of Technology, P.O.Box 16315-1618, Tehran, Iran. e-mail: [email protected] e-mail: [email protected]
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Abstract

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Let $v$ be a henselian valuation of any rank of a field $K$ and let $\bar{v}$ be the unique extension of $v$ to a fixed algebraic closure $\overline{K}$ of $K$. In 2005, we studied properties of those pairs $\left( \theta ,\,\alpha \right)$ of elements of $\overline{K}$ with $\left[ K\left( \theta \right):K \right]\,>\,\left[ K\left( \alpha \right):K \right]$ where $\alpha $ is an element of smallest degree over $K$ such that

$$\bar{v}\left( \theta \,-\,\alpha \right)\,=\,\sup \left\{ \bar{v}\left( \theta \,-\,\beta \right)\,|\,\beta \,\in \,\bar{K},\,\left[ K\left( \beta \right):K \right]\,<\,\left[ K\left( \theta \right):K \right] \right\}\,.$$

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

12J1012J2512E05valued fieldsnon-Archimedean valued fieldsirreducible polynomials
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 225 - 232
Copyright
Copyright © Canadian Mathematical Society 2015

References

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