Eisenstein's Irreducibility Criterion

Eisenstein's criterion provides a simple sufficient condition for irreducibility of polynomials over the integers (or more generally, over a UFD).

Statement

Theorem6.7Eisenstein's irreducibility criterion

Let $R$ be a UFD with field of fractions $K$ , and let $f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0 \in R[x]$ with $n \geq 1$ . If there exists a prime $p \in R$ such that:

$p \nmid a_n$ (the leading coefficient),
$p \mid a_i$ for all $0 \leq i < n$ ,
$p^2 \nmid a_0$ (the constant term),

then $f$ is irreducible in $K[x]$ . If additionally $f$ is primitive (the $a_i$ have no common factor), then $f$ is irreducible in $R[x]$ .

Proof

By Gauss's lemma, it suffices to show $f$ is irreducible in $R[x]$ (assuming $f$ is primitive).

Suppose $f = gh$ in $R[x]$ with $g = b_r x^r + \cdots + b_0$ and $h = c_s x^s + \cdots + c_0$ , where $r + s = n$ , $r, s \geq 1$ .

Since $a_0 = b_0 c_0$ and $p \mid a_0$ but $p^2 \nmid a_0$ : exactly one of $b_0, c_0$ is divisible by $p$ . WLOG $p \mid b_0$ and $p \nmid c_0$ .

Since $a_n = b_r c_s$ and $p \nmid a_n$ : $p \nmid b_r$ .

Consider the smallest index $k$ with $p \nmid b_k$ (so $1 \leq k \leq r$ and $p \mid b_0, \ldots, b_{k-1}$ ). The coefficient $a_k = b_k c_0 + b_{k-1}c_1 + \cdots + b_0 c_k$ . Since $p \mid a_k$ (as $k < n$ if $k < r$ ... wait, we need $k \leq r \leq n-1$ , so $k < n$ ) and $p \mid b_j$ for $j < k$ : $p \mid b_k c_0$ . Since $p \nmid c_0$ : $p \mid b_k$ , contradicting the choice of $k$ .

Therefore no such factorization exists, and $f$ is irreducible. $\blacksquare$

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Applications

ExampleApplications of Eisenstein's criterion

Cyclotomic polynomials: $\Phi_p(x) = x^{p-1} + x^{p-2} + \cdots + x + 1$ is irreducible over $\mathbb{Q}$ for prime $p$ . Apply the substitution $x \mapsto x + 1$ : $\Phi_p(x+1) = \binom{p}{1}x^{p-2} + \cdots + \binom{p}{p-2}x + p$ . Eisenstein with $p$ gives irreducibility.
$x^n - 2$ is irreducible over $\mathbb{Q}$ for all $n \geq 1$ (Eisenstein with $p = 2$ ).
$x^4 + 1$ is irreducible over $\mathbb{Q}$ (use $x \mapsto x+1$ : $(x+1)^4+1 = x^4+4x^3+6x^2+4x+2$ , Eisenstein with $p = 2$ ).

RemarkLimitations

Eisenstein's criterion is sufficient but not necessary. The polynomial $x^4 + 1$ is irreducible over $\mathbb{Q}$ but no direct application of Eisenstein works without a substitution. More generally, many irreducible polynomials (like $x^4 + x^3 + x^2 + x + 1 = \Phi_5(x)$ ) require substitutions to apply Eisenstein.