Irreducibility of polynomials

In many contexts, it is important to know whether a polynomial is irreducible. In algebraic number theory, the properties of an algebraic number $\alpha$ are determined by its minimal polynomial, the irreducible monic polynomial of least degree having $\alpha$ as a root. In Galois theory, it is important to know whether a field extension arises from an irreducible polynomial. In algebraic geometry, one is interested in varieties arising from the zero set of a set of multivariate polynomials, and if some of the polynomials are reducible, the variety can be represented in simpler terms. There are many more examples as well.

By the fundamental theorem of algebra, polynomials with complex coefficients always factor into linear factors in $\mathbb{C}[x].$ From this it follows that real polynomials factor into at most quadratic factors in $\mathbb{R}[x]$. However, in $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$, irreducibility is much more interesting. By Gauss' lemma, irreducibility in both of them is the same thing, so we consider only $\mathbb{Z}[x]$ (every polynomial in $\mathbb{Q}[x]$ becomes a polynomial with integer coefficients when multiplied by a suitable integer).

In what follows, by polynomials we mean polynomials with integer coefficients in one variable and irreducibility is considered in $\mathbb{Z}[x]$, unless otherwise noted. In this post, we present a few methods to determine whether a polynomial is irreducible. For some reason, it seems that the Eisenstein criterion is almost the only irreducibility criterion that is truly well-known, even though it fails for many polynomials.