Eisenstein's Criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers: if some prime divides every coefficient except the leading one, does not divide the leading coefficient, and its square does not divide the constant term, the polynomial cannot be factored into lower-degree polynomials with rational coefficients. Named for Ferdinand Eisenstein, it is a standard tool for proving irreducibility without directly searching for factorizations.
Facts
StatementIf there exists a prime number p such that p divides every coefficient of the polynomial except the leading one, p does not divide the leading coefficient, and p squared does not divide the constant term, then the polynomial is irreducible over the rational numbers. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Eisenstein's criterion - Wikipedia
Criterion section
If there exists a prime number p such that the following three conditions all apply: p divides each ai for 0 ≤ i < n, p does not divide an, and p² does not divide a₀, then Q is irreducible over the rational numbers.
History section
Subsequently, Eisenstein published a somewhat different version in 1850, also in Crelle's Journal.
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