On coefficient valuations of Eisenstein polynomials
Matthias Künzer1; Eduard Wirsing2
1 Universität Ulm Abt. Reine Mathematik D-89069 Ulm, Allemagne
2 Universität Ulm Fak. f. Mathematik D-89069 Ulm, Allemagne
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 801-823.

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.

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.

DOI: 10.5802/jtnb.522
Matthias Künzer 1; Eduard Wirsing 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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Matthias Künzer; Eduard Wirsing. On coefficient valuations of Eisenstein polynomials. Journal de théorie des nombres de Bordeaux, Volume 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 | Zbl

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

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

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

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

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

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

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

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

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

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