Eisenstein's criterion gives a sufficient condition for a polynomial in $x$ over $\mathbb{Z}$ to be irreducible over $\mathbb{Q}.$ > [!NOTE] Theorem (Schönemann-Eisentein Criterion, Eisentein's Criterion) > Let $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{0}\in \mathbb{Z}[x]$ be a polynomial in the [[Ring of Polynomial Forms|ring of polynomials]] in $x$ over the [[Integers|integers]]. If there exists a [[Prime numbers|prime]] $p$ such that $p \mid a_{i} \iff i\neq n$ and $p^{2}\not \mid a_{0}$ then $f$ is [[Irreducible Polynomial|irreducible]] over $\mathbb{Z}$ (and thus irreducible over $\mathbb{Q}$ by [[Polynomial which is Irreducible over Integers is Irreducible over Rationals|Gauss's lemma]]). > **Note** that $p\mid a_{i}$ signifies that $p$ is a [[Divisibility in Integers|divisor]] of $a_{i}.$ **Proof**: .... > [!Example] Example (Eisenstein's criterion is sufficient) > $x^{3}+2x+2$ is irreducible over $\mathbb{Q}$ by Eisenstein's criterion since $2$ divides every coefficient except $1$ and $2^{2}$ does not divide the constant term. > [!Example] Example (Eisenstein's criterion is not necessary) > $x^{3}+2x+4$ is irreducible over $\mathbb{Q}$ but doesn't satisfy Eisenstein's criterion. # Applications **Consequences**: If $p$ is a prime number then [[Irreducibility of the pth Cyclotomic Polynomial|pth cyclotomic polynomial is irreducible over the rationals]].