Eisenstein's criterion

Statement

Let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x]$ with $n \ge 1$, and let $p$ be a prime number with these three properties.

1. $p$ divides $a_i$ for each $i$ with $0 \le i < n$.
2. $p$ does not divide $a_n$.
3. $p^2$ does not divide $a_0$.

Then $f$ is irreducible in $\mathbb{Q}[x]$.

Remarks

If $f$ is also primitive, then $f$ is irreducible in $\mathbb{Z}[x]$, by Gauss's lemma. For example, $x^n - 2$ is irreducible over $\mathbb{Q}$ for each $n \ge 1$, with $p = 2$. A change of variable can help: $f(x + 1)$ is irreducible if and only if $f(x)$ is. This proves that $x^{p-1} + \dots + x + 1$ is irreducible for $p$ prime. The criterion holds more generally for a prime ideal $P$ of an integral domain $R$, with $P^2$ in place of $p^2$.

History

Theodor Schönemann published a version of the criterion in 1846. Gotthold Eisenstein published a different version in 1850. Both papers appeared in Crelle's Journal.

Explanations

L1 human

The criterion is a quick test of irreducibility over the rationals. Find one prime that divides all the coefficients except the leading one, and whose square does not divide the constant term. For $x^5 + 6x + 3$, the prime $3$ works, so the polynomial is irreducible.