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

Soit $p\ge 3$ un nombre premier et soient $n>m\ge 1$. Soit ${\pi }_{n}$ la norme de ${\zeta }_{{p}^{n}}-1$ sous ${C}_{p-1}$. Ainsi ${ℤ}_{\left(p\right)}\left[{\pi }_{n}\right]|{ℤ}_{\left(p\right)}$ est une extension purement ramifiée d’anneaux de valuation discrète de degré ${p}^{n-1}$. Le polynôme minimal de ${\pi }_{n}$ sur $ℚ\left({\pi }_{m}\right)$ est un polynôme de Eisenstein ; nous donnons des bornes inférieures pour les ${\pi }_{m}$-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 $p\ge 3$ be a prime, let $n>m\ge 1$. Let ${\pi }_{n}$ be the norm of ${\zeta }_{{p}^{n}}-1$ under ${C}_{p-1}$, so that ${ℤ}_{\left(p\right)}\left[{\pi }_{n}\right]|{ℤ}_{\left(p\right)}$ is a purely ramified extension of discrete valuation rings of degree ${p}^{n-1}$. The minimal polynomial of ${\pi }_{n}$ over $ℚ\left({\pi }_{m}\right)$ is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at ${\pi }_{m}$. The function field analogue, as introduced by Carlitz and Hayes, is studied as well.

Publié le :
DOI : https://doi.org/10.5802/jtnb.522
@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/

[1] L. Carlitz, A class of polynomials. Trans. Am. Math. Soc. 43 (2) (1938), 167–182. | MR 1501937 | Zbl 0018.19806

[2] R. Dvornicich, U. Zannier, Sums of roots of unity vanishing modulo a prime. Arch. Math. 79 (2002), 104–108. | MR 1925376 | Zbl 1007.11062

[3] D. Goss, The arithmetic of function fields. II. The “cyclotomic” theory. J. Alg. 81 (1) (1983), 107–149. | MR 696130 | Zbl 0516.12010

[4] D. Goss, Basic structures of function field arithmetic. Springer, 1996. | MR 1423131 | Zbl 0892.11021

[5] D. R. Hayes, Explicit class field theory for rational function fields. Trans. Am. Math. Soc. 189 (2) (1974), 77–91. | MR 330106 | Zbl 0292.12018

[6] T. Y. Lam, K. H. Leung, On Vanishing Sums of Roots of Unity. J. Alg. 224 (2000), 91–109. | MR 1736695 | Zbl 01423896

[7] J. Neukirch, Algebraische Zahlentheorie. Springer, 1992. | Zbl 0747.11001

[8] M. Rosen, Number Theory in Function Fields. Springer GTM 210, 2000. | MR 1876657 | Zbl 1043.11079

[9] J. P. Serre, Corps Locaux. Hermann, 1968. | MR 354618 | Zbl 0137.02601

[10] H. Weber, M. Künzer, Some additive galois cohomology rings. Arxiv math.NT/0102048, to appear in Comm. Alg., 2004. | MR 2188320 | Zbl 02248211