On coefficient valuations of Eisenstein polynomials
Matthias Künzer ; Eduard Wirsing
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 801-823.

Soit p≥3 un nombre premier et soient n>m≥1. Soit π n la norme de ζ p n -1 sous C p-1 . Ainsi ℤ (p) [π n ]|ℤ (p) est une extension purement ramifiée d’anneaux de valuation discrète de degré p n-1 . Le polynôme minimal de π n sur ℚ(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 p≥3 be a prime, let n>m≥1. Let π n be the norm of ζ p n -1 under C p-1 , so that ℤ (p) [π n ]|ℤ (p) is a purely ramified extension of discrete valuation rings of degree p n-1 . The minimal polynomial of π n over ℚ(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.

@article{JTNB_2005__17_3_801_0,
     author = {Matthias K\"unzer and Eduard Wirsing},
     title = {On coefficient valuations of Eisenstein polynomials},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {801--823},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {3},
     year = {2005},
     doi = {10.5802/jtnb.522},
     zbl = {05016589},
     mrnumber = {2212127},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.522/}
}
Matthias Künzer; Eduard Wirsing. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.522/

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