Eisenstein's irreducibility criterion is a method for proving that a polynomial with integer coefficients is irreducible (that is, cannot be written as a product of two polynomials of smaller degree with integer coefficients). Due to its specific requirements, it is not generally applicable to most polynomials, but it is useful for exhibiting examples of carefully chosen polynomials which are irreducible.
Eisenstein's Irreducibility Criterion
Let be a polynomial with integer coefficients. Suppose that there exists a prime , such that
Then is irreducible over the integers.
It is a corollary of Gauss's lemma that is also irreducible over the rational numbers.
Suppose not, then
with and . Take least such that (where we allow the case and ), and least such that (where we allow the case and ). Now the coefficient of the product is
and so is a sum of and terms divisible by . As , . But so cannot divide both and . Thus, either or so . So we contradict .
Thus, by contradiction, Eisenstein's irreducibility criterion holds true.
Show that there is an irreducible polynomial with integer coefficients of degree for any positive integer
The polynomial is irreducible by Eisenstein's criterion (with
Let be a prime number. Show that is irreducible.
does not satisfy the conditions of Eisenstein's criterion. But does, because is divisible by for , and is not divisible by . So is irreducible. But if factored as , then so would . So is irreducible as well.