Eisenstein's criterion /

In mathematics, Eisenstein's criterion gives an easily checked sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers. The result is also known as the Schönemann–Eisenstein theorem; although this name is rarely used nowadays, it was common in the early 20th century. Suppose we have the following polynomial with integer coefficients. If there exists a prime number p such that the following three conditions all apply: then Q is irreducible over the rational numbers. If in addition Q is primitive, in other words if no other prime number divides all coefficients of Q at once, then Q is also irreducible over the integers. In case Q is not primitive, it can be made so by dividing by the greatest common divisor of its coefficients (the contents of Q), which does not change whether it is reducible or not over the rational numbers, and will not invalidate the hypotheses of the criterion for any prime number for which they are satisfied. This criterion is certainly not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but does allow in certain important particular cases to prove