Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T21:30:39.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Factorization over local fields and the irreducibility of generalized difference polynomials

Part of: General field theory

Published online by Cambridge University Press:  26 February 2010

S. D. Cohen ,
A. Movahhedi  and
A. Salinier
S. D. Cohen
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland.
A. Movahhedi
Affiliation:
UPRES A 6090 CNRS, Faculté des Sciences, 123 Avenue Albert Thomas, 87060 Limoges Cedex, France.
A. Salinier
Affiliation:
UPRES A 6090 CNRS, Faculté des Sciences, 123 Avenue Albert Thomas, 87060 Limoges Cedex, France.
Get access

Extract

Let E be a local field, i.e., a field which is complete with respect to a rank one discrete valuation υ (we do not require any finiteness condition on the residue class field of E). Let f(X) be a polynomial in one variable, with coefficients in E. It is well known [4, 6, 9, 11, 13] that the Newton polygon method allows us to gather information about the factorization of f(X). This method consists of attaching to each side S of a Newton polygon of f(X) a factor (not necessarily irreducible) of f(X), the degree of which is the length of the horizontal projection of S.

MSC classification

Secondary: 12E05: Polynomials (irreducibility, etc.)
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 173 - 196
Copyright
Copyright © University College London 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abhyankar, S. S. and Rubel, L. A.. Every difference polynomial has a connected zero-set. J. Indian Math. Soc., 43 (1979), 69–78.Google Scholar
2.Alexandra, V., Popescu, N. and Zaharescu, A.. A theorem of characterization of residual transcendental extensions of a valuation. J. Math. Kyoto Univ., 28 (1988), 579–592.Google Scholar
3.Alexandra, V., Popescu, N. and Zaharescu, A.. Minimal pairs of definition of a residual transcendental extension of a valuation. J. Math. Kyoto Univ., 30 (1990), 207–225.Google Scholar
4.Amice, Y.. Les nombres p-adiques Collection Sup “Le mathematicien”, 14 (Presses Universitaires de France, Paris, 1975).Google Scholar
5.Angermüller, G.. A generalization of Ehrenfeucht's irreducibility criterion. J. Number Theory, 36 (1990), 80–84.CrossRefGoogle Scholar
6.Artin, E.. Algebraic Numbers and Algebraic Functions. Notes on Mathematics and Its Applications (Gordon and Breach. New York, London, Paris, 1967).Google Scholar
7.Bosch, S., Güntzer, U. and Remmert, R.. Non-Archimedean Analysis, Grundlehren der mathematischen Wissenschaften, 261 (Springer-Verlag, Berlin, 1984).CrossRefGoogle Scholar
8.Bourbaki, N.. Algèbre Commutative, chap. 6 (Hermann, Paris, 1964).Google Scholar
9.Cassels, J. W. S.. Local Fields, London Mathematical Society Student Texts, 3 (Cambridge University Press, Cambridge, 1986).Google Scholar
10.Cohen, S. D., Movahhedi, A. and Salinier, A.. Double transitivity of Galois groups of trinomials. Acta Arith., 82 (1997), 1–15.CrossRefGoogle Scholar
11.Gouvea, F. Q.. P-adic numbers. An introduction, Universitext (Springer-Verlag, Berlin, 1993).CrossRefGoogle Scholar
12.Khanduja, S. and Saha, J.. On a generalization of Eisenstein's irreducibility criterion. Mathematika, 44 (1997), 37–41.CrossRefGoogle Scholar
13.Koblitz, N.. P-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics, 58 (Springer-Verlag, New York, Heidelberg, Berlin, 1977).CrossRefGoogle Scholar
14.Lazard, M.. Polygone de Newton et théorème de préparation, Séminaire Dubreil, 26e année, 1972/1973, no. 15, 4 p.Google Scholar
15.Matignon, M. and Ohm, J.. A structure theorem for simple transcendental extensions of valued fields. Proc. Amer. Math. Soc., 104 (1988), 392–402.CrossRefGoogle Scholar
16.Matignon, M. and Reversat, M.. Sous-corps fermés d'un corps value. J. Algebra, 90 (1984), 491–515.CrossRefGoogle Scholar
17.Montes, J. and Nart, E.. On a Theorem of Ore. J. Algebra, 146 (1992), 318–334.CrossRefGoogle Scholar
18.Narkiewicz, W.. Elementary and Analytic Theory of Algebraic Numbers, Second edition (Springer-Verlag, Berlin; PWN-Polish Scientific Publishers, Warsaw, 1990).Google Scholar
19.Ore, O.. Newtonsche Polygone in der Theorie der algebraischen Körper. Math. Ann., 99 (1928), 84–117.CrossRefGoogle Scholar
20.Panaitopol, L. and Stefanescu, D.. On the generalized difference polynomials. Pacific J. Math., 143 (1990), 341–348.CrossRefGoogle Scholar
21.Popescu, N. and Zaharescu, A.. On the structure of the irreducible polynomials over local fields. J. Number Theory, 52 (1995), 98–118.CrossRefGoogle Scholar
22.Robba, P. and Christol, G.. Equations differentielles p-adiques (Hermann, Paris, 1994).Google Scholar
23.Rockafellar, R. T.. Convex Analysis, Second Printing (Princeton University Press, Princeton, New Jersey, 1972).Google Scholar
24.Rubel, L. A., Schinzel, A. and Tverberg, H.. On difference polynomials and hereditary irreducible polynomials. J. Number Theory, 12 (1980), 230–235.CrossRefGoogle Scholar