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

Bibliographie

[1] [1] Aghigh, K. and Khanduja, S. K., On the main invariant of elements algebraic over a Henselian valued field. Proc. Edinb. Math. Soc. 45 (2002), no. 1, 219–227. Google Scholar

[2] [2] Aghigh, K. and Khanduja, S. K., On chains associated with elements algebraic over a Henselian valued field. Algebra Colloq. 12 (2005), no. 4, 607–616. http://dx.doi.Org/10.1142/S100538670500057X Google Scholar

[3] [3] Alexandru, V., Popescu, N., and Zaharescu, A., A theorem of characterization of residual transcendental extensions of a valuation. J. Math. Kyoto Univ. 28 (1988), no. 4, 579–592. Google Scholar

[4] [4] Alexandru, V., Popescu, N., and Zaharescu, A., Minimal pairs of definition of a residual transcendental extension of a valuation. J. Math. Kyoto Univ. 30 (1990), no. 2, 207–225. Google Scholar

[5] [5] Bhatia, S. and Khanduja, S. K., On extensions generated by roots of lifting polynomials. Mathematika 49 (2002), no. 1–2,107–118. http://dx.doi.Org/10.1112/S0025579300016107 Google Scholar

[6] [6] Bishoni, A. and Khanduja, S. K., On Eisenstein-Dumas and generalized Schonemann polynomials. Comm. Algebra 38 (2010), no. 9, 3163–3173. http://dx.doi.Org/10.1080/00927870903164669 Google Scholar

[7] [7] Bishoni, A., Kumar, S., and Khanduja, S. K., On liftings of powers of irreducible polynomials. J. Algebra Appl. 12 (2013), no. 5,1250222. http://dx.doi.Org/10.1142/S0219498812502222 Google Scholar

[8] [8] Brown, R. and Merzel, J. L., Invariants of defectless irreducible polynomials. J. Algebra Appl. 9 (2010), no. 4, 603–631. http://dx.doi.Org/1 0.1142/S021 949881 0004130 Google Scholar

[9] [9] Khanduja, S. K., On valuations ofK(x). Proc. Edinburgh Math. Soc. 35 (1992), no. 3, 419–426. Google Scholar | DOI

[10] [10] Khanduja, S. K. and Kumar, S., On prolongation of valuations via Newton polygons and liftings of polynomials. J. Pure Appl. Algebra 216 (2012), no. 12, 2648–2656. http://dx.doi.Org/10.1016/j.jpaa.2012.03.034 Google Scholar

[11] [11] Khanduja, S. K. and Saha, J., On a generalization of Eisenstein's irreducibility criterion. Mathematika 44 (1997), no. 1, 37–41. http://dx.doi.Org/10.1112/S0025579300011931 Google Scholar

[12] [12] Khanduja, S. K. and Saha, J., A generalized fundamental principle. Mathematika 46 (1999), no. 1, 83–92. http://dx.doi.Org/10.1112/S0025579300007580 Google Scholar

[13] [13] Popescu, N. and Zaharescu, A., On the structure of the irreducible polynomials over local fields. J. Number Theory 52 (1995), no. 1, 98–118. http://dx.doi.Org/10.1006/jnth.1995.1058 Google Scholar