Eisenstein's Irreducibility Criterion

Eisenstein's irreducibility criterion is a sufficient condition assuring that an integer polynomial  is irreducible in the polynomial ring .

The polynomial

where  for all  and  (which means that the degree of  is ) is irreducible if some prime number  divides all coefficients , ..., , but not the leading coefficient  and, moreover,  does not divide the constant term .

This is only a sufficient, and by no means a necessary condition. For example, the polynomial  is irreducible, but does not fulfil the above property, since no prime number divides 1. However, substituting  for  produces the polynomial , which does fulfill the Eisenstein criterion (with ) and shows the polynomial is irreducible.

REFERENCES:

Childs, L. A Concrete Introduction to Higher Algebra. New York: Springer-Verlag, pp. 169-172, 1979.

Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, pp. 160-161, 1975.