One of the irreducible criteria is well-known for polynomials with integers coefficients and also holds naturally for polynomials of is Eisenstein's irreducible criterion that states
Criterion 1 (Eisenstein's irreducible criterion).
Let . If there exists a prime number satisfying the following three conditions:
- divides each for . - does not divide , and - does not divide .
Then the polynomial is irreducible.
We have another criterion, which is even easier to check.
Criterion 2: Let . Suppose that and is a prime number, then is irreducible.
Proof
Indeed, suppose that for some . Note that , and since is prime, we know that either or . WLOG, assume that , then for some . Note that if , then , and then , which contradicts the hypothesis. Therefore , and this implies that is irreducible.