On a generalization of Eisenstein's irreducibility criterion

Abstract

Let ν be a valuation of any rank of a field K with value group G_ν and f(X)= X^m + a_lX^m−1 + … + a_m be a polynomial over K. In this paper, it is shown that if (ν(a_i)/i)≥(ν(a_m)/m)>0 for l≤i≤m, and there does not exist any integer r>1 dividing m such that ν(a_m)/r∈G_ν, 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]).