nLab
irreducible polynomial

A non-zero polynomial $f$ with coefficients in a field $k$ is irreducible if when written as the product $gh$ of two polynomials, one of $g$ or $h$ is a constant (and necessarily non-zero) polynomial. Equivalently, a polynomial $f$ is irreducible if the ideal it generates is a maximal ideal of the polynomial ring $k[x]$.

In other words, a polynomial $f$ is irreducible if it is an irreducible element? of $k[x]$ as an integral domain.

Notice that under this definition, the zero polynomial is not considered to be irreducible. An alternative definition, which applies to the case of coefficients in a commutative ring $R$, is that a polynomial $f$ is irreducible if, whenever $f$ divides $gh$, either $f$ divides $g$ or $f$ divides $h$. Under this definition, a polynomial is irreducible if it generates a prime ideal in $R[x]$, and the zero polynomial is irreducible if $R$ is an integral domain.