ON A GENERALIZATION OF EISENSTEINS IRREDUCIBILITY CRITERION

Let v be a valuation of any rank of a field K with value group G(v) an d f(X) = X-m + a(1)X(m-1) +...+ a(m) be a polynomial over K. In this p aper, it is shown that if (v(a(i))/i) greater than or equal to (v(a(m) )/m) > 0 for 1 less than or equal to i less than or equal to m, and th ere does not exist any integer r>1 dividing m such that v(a(m))/r epsi lon G(v), 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 Ei senstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields.