A CLASSIFICATION OF EXTENSIONS GENERATED BY A ROOT OF AN EISENSTEIN-DUMAS POLYNOMIAL

Nikseresht, َAzadeh

doi:10.22044/jas.2022.11808.1603

Document Type : Original Manuscript

Author

َAzadeh Nikseresht

Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran.

https://doi.org/10.22044/jas.2022.11808.1603

Abstract

It is known that for a discrete valuation v of a field K with value group Z, an valued extension field (K′, v′) of (K, v) is generated by a root of an Eisenstein polynomial with respect to v having coefficients in K if and only if the extension (K′, v′)/(K, v) is totally ramified. The aim of this paper is to present the analogue of this result for valued field extensions generated by a root of an Eisenstein-Dumas polynomial with respect to a more general valuation (which is not necessarily discrete). This leads to classify such algebraic extensions of valued fields.

Keywords

References

1. R. Brown, Roots of generalized Schönemann polynomials in henselian extension fields, Indian J. Pure Appl. Math., 39 (2008), 403–410.

2. G. Dumas, Sur quelques cas d’irreductibilité des polynômes á coefficients rationnels, Journal de Math. Pure et Appl., 2 (1906), 191–258.

3. G. Eisenstein, Über die Irreductibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt, J. Reine Angew. Math., 39 (1850), 160–182.

4. A. J. Engler and A. Prestel, Valued Fields, Springer-Verlag, Berlin, 2005.

5. S. K. Khanduja and J. Saha, On a generalization of Eisenstein’s irreducibility criterion, Mathematika, 44(1) (1997), 37–41.

6. W. Krull, Allgemeine bewertungstheorie, Journ. f. d. reine u. angewandte Math., 167 (1932), 160–196.

7. F.-V. Kuhlmann, Valuation theory of fields, abelian groups and modules, monograph in preparation, preliminary versions of several chapters are available on the web site http //: math.usask.ca/ fvk/Fvkbook.htm.

8. J. Kürschák, Irreduzible formen, J. Reine Angew. Math., 152 (1923), 180–191.

9. J. Kürschák, Über Limesbildung und allgemeine Körpertheorie, Proceedings of

the 5th International Congress of Mathematicians Cambridge 1912, 1 (1913), 285–289.

10. P. Ribenboim, The Theory of Classical Valuations, Springer Monographs in Mathematics, Springer-Verlag, New York, 1999.

Journal of Algebraic Systems

A CLASSIFICATION OF EXTENSIONS GENERATED BY A ROOT OF AN EISENSTEIN-DUMAS POLYNOMIAL

References

References

Volume 11, Issue 2
January 2024
Pages 83-91

A CLASSIFICATION OF EXTENSIONS GENERATED BY A ROOT OF AN EISENSTEIN-DUMAS POLYNOMIAL

References

References

Volume 11, Issue 2January 2024Pages 83-91

Volume 11, Issue 2
January 2024
Pages 83-91