Eisenstein-Dumas criterion for irreducibility and linear transformations
by
Martin Juras
North Dakota State University
Coauthors: This is my second submission (of the same talk). I feel that the abstract that I submitted last time was too short. Please, be assured this is my last submission. I appologize for any inconveniences this may cause. Thank you.

Eisenstein-Dumas criterion, that first appeared in a paper by G.Dumas in 1906, is a generalization of Eisenstein's criterion for irreducibility of polynomials. We study the following question: Let \Cal F be a quotient field of a UFD \Cal R. Given polynomial A(x) in \Cal F[x], is there a linear transformation x --> ax+b a, b in \Cal F such that A(ax+b) satisfies Eisenstein-Dumas criterion for irreducibility with some prime p in \Cal R? We partially solve this problem. We show for instance, that if \char \Cal R does not divide the degree n of the polynomial A(x)=a_nxⁿ+a_n-1x^n-1+ ... and if p in \Cal R is a prime, such that p does not divide n=1+1+ ... +1 in \Cal R, then the polynomial A(ax+b) satisfies Eisenstein-Dumas criterion with p for some a and b iff the polynomial [`A](x)=A(x-\fraca_n-1na_n) satisfies Eisenstein-Dumas criterion with p. We find a complete solution to the problem whether or not for a given polynomial A(x) with integer coefficients there are integers a, b, an c, ab =/= 0, such that the polynomial bⁿ A(ax+\fraccb) satisfies classical Eisenstein's criterion with some prime p. We also discuss projective and polynomial transformations. We give an example of an irreducible polynomial A(x) of degree 4 with integer coefficients such that for no rational numbers a, b, c, and d, the polynomial (cx+d)⁴A(\fracax+bcx+d) satisfies Eisenstein-Dumas criterion.

Date received: February 20, 2004