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Date: 11-3-2019
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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.
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