On coefficient valuations of Eisenstein polynomials
Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 801-823.

Soit p3 un nombre premier et soient n>m1. Soit π n la norme de ζ p n -1 sous C p-1 . Ainsi (p) [π n ]| (p) est une extension purement ramifiée d’anneaux de valuation discrète de degré p n-1 . Le polynôme minimal de π n sur (π m ) est un polynôme de Eisenstein ; nous donnons des bornes inférieures pour les π m -valuations de ses coefficients. L’analogue dans le cas d’un corps de fonctions, comme introduit par Carlitz et Hayes, est etudié de même.

Let p3 be a prime, let n>m1. Let π n be the norm of ζ p n -1 under C p-1 , so that (p) [π n ]| (p) is a purely ramified extension of discrete valuation rings of degree p n-1 . The minimal polynomial of π n over (π m ) is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at π m . The function field analogue, as introduced by Carlitz and Hayes, is studied as well.

DOI : 10.5802/jtnb.522
Künzer, Matthias 1 ; Wirsing, Eduard 2

1 Universität Ulm Abt. Reine Mathematik D-89069 Ulm, Allemagne
2 Universität Ulm Fak. f. Mathematik D-89069 Ulm, Allemagne
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Künzer, Matthias; Wirsing, Eduard. On coefficient valuations of Eisenstein polynomials. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 801-823. doi : 10.5802/jtnb.522. http://www.numdam.org/articles/10.5802/jtnb.522/

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