Perturbing Eisenstein polynomials over local fields
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 681-694.

Let K be a local field whose residue field has characteristic p and let L/K be a finite separable totally ramified extension. Let π L be a uniformizer for L and let f(X) be the minimum polynomial for π L over K. Suppose π ˜ L is another uniformizer for L such that π ˜ L π L +rπ L +1 (modπ L +2 ) for some 1 and r𝒪 K . Let f ˜(X) be the minimum polynomial for π ˜ L over K. In this paper we give congruences for the coefficients of f ˜(X) in terms of , r, and the coefficients of f(X). These congruences improve work of Krasner [8].

Soit K un corps local de caractéristique résiduelle p et soit L/K une extension séparable finie totalement ramifiée. Soit π L une uniformisante de L, de polynôme minimal f(X) sur K. Supposons que π ˜ L est une autre uniformisante de L telle que π ˜ L π L +rπ L +1 (modπ L +2 ) pour certains 1 et r𝒪 K . Soit f ˜(X) le polynôme minimal de π ˜ L sur K. Dans cet article nous donnons des congruences pour les coefficients de f ˜(X) en termes de , r, et les coefficients de f(X). Ces congruences améliorent le travail de Krasner [8].

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1045
Classification: 11S15, 11S05
Keywords: local fields, Eisenstein polynomials, symmetric polynomials, indices of inseparability, digraphs
Keating, Kevin 1

1 Department of Mathematics University of Florida Gainesville, FL 32611, USA
@article{JTNB_2018__30_2_681_0,
     author = {Keating, Kevin},
     title = {Perturbing {Eisenstein} polynomials over local fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {681--694},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1045},
     mrnumber = {3891333},
     zbl = {1441.11294},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1045/}
}
TY  - JOUR
AU  - Keating, Kevin
TI  - Perturbing Eisenstein polynomials over local fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 681
EP  - 694
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1045/
DO  - 10.5802/jtnb.1045
LA  - en
ID  - JTNB_2018__30_2_681_0
ER  - 
%0 Journal Article
%A Keating, Kevin
%T Perturbing Eisenstein polynomials over local fields
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 681-694
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1045/
%R 10.5802/jtnb.1045
%G en
%F JTNB_2018__30_2_681_0
Keating, Kevin. Perturbing Eisenstein polynomials over local fields. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 681-694. doi : 10.5802/jtnb.1045. http://www.numdam.org/articles/10.5802/jtnb.1045/

[1] Deligne, Pierre Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0, Représentations des groups réductifs sur un corps local (Travaux en Cours), Hermann, 1984, pp. 119-157 | Zbl

[2] Fried, Michael Arithmetical properties of function fields II, The generalized Schur problem, Acta Arith., Volume 25 (1974), pp. 225-258 | DOI | MR | Zbl

[3] Fried, Michael; Mézard, Ariane Configuration spaces for wildly ramified covers, Arithmetic fundamental groups and noncommutative algebra (Berkeley, 1999) (Proceedings of Symposia in Pure Mathematics), Volume 70, American Mathematical Society, 2002, pp. 353-376 | DOI | MR | Zbl

[4] Heiermann, Volker De nouveaux invariants numériques pour les extensions totalement ramifiées de corps locaux, J. Number Theory, Volume 59 (1996) no. 1, pp. 159-202 | DOI | MR | Zbl

[5] Jones, John W.; Roberts, David P. Database of Local Fields (https://math.la.asu.edu/ jj/localfields/, Retrieved 31 December 2016) | Zbl

[6] Jones, John W.; Roberts, David P. A database of local fields, J. Symb. Comput., Volume 41 (2006) no. 1, pp. 80-97 | DOI | MR | Zbl

[7] Keating, Kevin Extensions of local fields and elementary symmetric polynomials, J. Théor. Nombres Bordx, Volume 30 (2018) no. 2, pp. 431-448 | DOI | MR | Zbl

[8] Krasner, Marc Sur la primitivité des corps 𝔓-adiques, Mathematica, Volume 13 (1937), pp. 72-191 | Zbl

[9] Kulikauskas, Andrius; Remmel, Jeffrey Lyndon words and transition matrices between elementary, homogeneous and monomial symmetric functions, Electron. J. Comb., Volume 13 (2006) no. 1, R18, 30 pages (Art. ID. R18, 30 p.) | MR | Zbl

[10] Serre, Jean-Pierre Corps Locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, 8, Hermann, 1962, 243 pages | Zbl

[11] Stanley, Richard P. Enumerative combinatorics, Volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999, xii+581 pages | Zbl

Cited by Sources: