Eisenstein-type irreducibility criteria. (English) Zbl 0938.12005

Anderson, David F. (ed.) et al., Zero-dimensional commutative rings. Proceedings of the 1994 John H. Barrett memorial lectures and conference on commutative ring theory, Knoxville, TN, USA, April 7-9, 1994. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 171, 307-329 (1995).
The paper firstly gives a short historical review of Eisenstein–type irreducibility criteria for univariate polynomials. Then, the author proposes to resurrect the use of Newton polygons to deal with polynomial factorization and irreducibility. He gives new detailed proofs of some old irreducibility criteria, mainly due to G. Dumas [J. Math. Pures Appl. (6) 2, 191-258 (1906; JFM 37.0096.01)] and O. Perron [Math. Ann. 60, 448-458 (1905; JFM 36.0123.01)], providing several examples and some interesting historical and technical observations. Finally, some new results are obtained and they are used to prove the irreducibility of some trinomials of the form \(X^n+aX+b\), where \(a\) and \(b\) are equal to \(1\) or \(2\); furthermore, it is shown that the use of Newton polygons could be very useful in testing irreducibility of a polynomial \(f\) when one knows by other means that \(f\) has no irreducible quadratic (or cubic, etc.) factors.
For the entire collection see [Zbl 0872.00033].


12Y05 Computational aspects of field theory and polynomials (MSC2010)
12E05 Polynomials in general fields (irreducibility, etc.)
12J25 Non-Archimedean valued fields
11R09 Polynomials (irreducibility, etc.)