Proving irreducibility of polynomials over the rational numbers.

Question

Can you provide me with some (easy to use) lemmas and criteria (besides Eisenstein's) to prove that a given polynomial is irreducible (if it is) over the rational numbers?

Answer 1

Cohn's irreducibility criterion: if the coefficients "spell out" a prime number in any base, then the polynomial is irreducible. More formally, let ${\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$ with ${\displaystyle 0\leq a_{k}\leq b-1}$ for some ${\displaystyle b\geq 2}$. If ${\displaystyle p(b)}$ is a prime number then ${\displaystyle p(x)}$ is irreducible in ${\displaystyle \mathbb {Z} [X]}$. Example: $\,x^3+6x^2+9x+3\,$ is irreducible because $\,1693_{10}\,$ is a prime.