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On a generalization of Eisenstein's irreducibility criterion

Part of: General field theory

Published online by Cambridge University Press:  26 February 2010

Sudesh K. Khanduja  and
Jayanti Saha
Sudesh K. Khanduja
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, India.
Jayanti Saha
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, India.
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Abstract

Let ν be a valuation of any rank of a field K with value group Gν and f(X)= Xm + alXm−1 + … + am be a polynomial over K. In this paper, it is shown that if (ν(ai)/i)≥(ν(am)/m)>0 for l≤i≤m, and there does not exist any integer r>1 dividing m such that ν(am)/r∈Gν, then f(X) is irreducible over K. It is derived as a special case of a more general result proved here. It generalizes the usual Eisenstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields, (cf. [Journal of Number Theory, 52 (1995), 98–118]).

MSC classification

Secondary: 12E05: Polynomials (irreducibility, etc.)
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 37 - 41
Copyright
Copyright © University College London 1997

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References

1.Alexandru, V., Popescu, N. and Zaharescu, A.. A theorem of characterization of residual transcendental extension of a valuation. J. Math. Kyoto Univ., 28 (1988), 579–592.Google Scholar
2.Popescu, L. and Popescu, N.. Sur la definition des prolongements residuals transcendents d'une valuation sur un corps K à K(x). Bull. Math. Sci. Math. R. S. Roumanie, 33 (81), No. 3 (1989), 257–264.Google Scholar
3.Popescu, N. and Zaharescu, A.. On the structure of irreducible polynomials over local fields. J. Number Theory, 52 (1995), 98–118.Google Scholar
4.Zariski, O. and Samuel, P.. Commutative Algebra, Vol. II (D. Van Nostrand Company Inc., Princeton, New Jersey).CrossRefGoogle Scholar