Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials

Aghigh, Kamal ; Nikseresht, Azadeh

Canadian mathematical bulletin, Tome 58 (2015) no. 2, pp. 225-232

Résumé

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.

Détail

DOI : 10.4153/CMB-2014-064-2

Mots-clés : 12J10, 12J25, 12E05, valued fields, non-Archimedean valued fields, irreducible polynomials