nLab irreducible polynomial

A non-zero polynomial $f$ with coefficients in a field $k$ is irreducible if when written as the product $g h$ 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 $g h$, 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.

If $L$ is a field extension of $k$ and $\alpha \in L$ is algebraic over $k$ (i.e., the smallest subextension containing $\alpha$, $k(\alpha)$, is finite-dimensional over $k$), then the $k$-algebra map $k[x] \to k(\alpha): x \mapsto \alpha$ has a non-trivial kernel which is a maximal ideal, which being principal is generated by an irreducible polynomial $f$. The unique monic polynomial generator is called the irreducible polynomial of $\alpha$.

Over a unique factorization domain eisenstein's criterion? determines irreducibility