Eisenstein’s criterion

Let $𝑅$ be an integral domain and $𝑓 (𝑥) = \sum 𝑛 𝑖 = 1 𝑎 𝑖 𝑥 𝑖 \in 𝑅 [𝑥]$ be a polynomial. For a prime ideal $𝔭 ⊴ 𝑅$ , we say $𝑓 (𝑥)$ is Eisenstein at $𝔭$ iff

$𝑎 𝑖 \in 𝔭$ for $1 \leq 𝑖 < 𝑛$ ;
$𝑎 𝑛 \notin 𝔭$ ;
$𝑎 0 \notin 𝔭 2$ .

If $𝑓 (𝑥)$ is Eisenstein at some prime ideal $𝔭$ , then $𝑓 (𝑥)$ cannot be written as the product of two non-constant polynomials in $𝑅 [𝑥]$ .¹ alg

Proof

proof

In particular, if $𝑅$ is a Unique factorization domain then $𝑓 (𝑥)$ is also irreducible in $(F r a c 𝑅) [𝑥]$ , by Gauß’s lemma.

develop | en | SemBr

2009. Algebra: Chapter 0, §V.5.4, p. 288 ↩

Eisenstein’s criterion

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Eisenstein’s criterion

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