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A Generalization of the Eisenstein–Dumas–Schönemann Irreducibility Criterion

Published online by Cambridge University Press:  31 January 2017

Bablesh Jhorar
Affiliation:
Department of Mathematics, Panjab University, Chandigarh-160014, India ([email protected])
Sudesh K. Khanduja*
Affiliation:
Indian Institute of Science Education and Research Mohali, Sector 81, SAS Nagar-140306, Punjab, India ([email protected])
*
*Corresponding author.

Abstract

In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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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), 579592.Google Scholar
2. Bhatia, S. and Khanduja, S. K., Difference polynomials and their generalizations, Mathematika 48 (2001), 293299.CrossRefGoogle Scholar
3. Bishnoi, A. and Khanduja, S. K., On Eisenstein–Dumas and generalized Schönemann polynomials, Commun. Alg. 38 (2010), 31633173.Google Scholar
4. Brown, R., Roots of generalized Schönemann polynomials in henselian extension fields, Indian J. Pure Appl. Math. 39(5) (2008), 403410.Google Scholar
5. Cox, D. A., Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first, Am. Math. Mon. 118(1) (2011), 321.CrossRefGoogle Scholar
6. Dumas, G., Sur quelques cas d’irréductibilité des polynômes á coefficients rationnels, J. Math. Pures Appl. 6(2) (1906), 191258.Google Scholar
7. Eisenstein, G., Über die Irreduztibilität und einige andere Eigenschaften der Gleichungen, von welcher die Theilung der ganzen Lemniscate abhängt, J. Reine Angew. Math. 39 (1850), 160179.Google Scholar
8. Engler, A. J. and Prestel, A., Valued fields (Springer, 2005).Google Scholar
9. Khanduja, S. K. and Khassa, R., A generalization of Eisenstein–Schönemann irreducibility criterion, Manuscr. Math. 134 (2011), 215224.CrossRefGoogle Scholar
10. Khanduja, S. K. and Kumar, M., Prolongations of valuations to finite extensions, Manuscr. Math. 131 (2010), 323334.Google Scholar
11. Khanduja, S. K. and Saha, J., On a generalization of Eisenstein's irreducibility criterion, Mathematika 44 (1997), 3741.Google Scholar
12. MacLane, S., The Schönemann–Eisenstein irreducibility criteria in terms of prime ideals, Trans. Am. Math. Soc. 43 (1938), 226239.Google Scholar
13. Schönemann, T., Von denjenigen Moduln, welche Potenzen von Primzahlen sind, J. Reine Angew. Math. 32 (1846), 93105.Google Scholar
14. Weintraub, S. H., A mild generalization of Eisenstein's criterion, Proc. Am. Math. Soc. 141(4) (2013), 11591160.CrossRefGoogle Scholar