Algebra and number theory · existing Mathlib declaration

Eisenstein Criterion

Choose a prime ideal P in an integral domain. If every coefficient below the leading term lies in P, the leading coefficient stays outside P, the constant coefficient stays outside P², and the polynomial is primitive with positive degree, then the polynomial cannot factor into two nonunits.

polynomial irreducibility
prime ideals
primitive polynomials
quotient rings
fraction fields
Eisenstein criterion

The theorem at a glance

Eisenstein Criterion at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

An ivory and dark-green theorem poster lists the exact prime-ideal, coefficient, positive-degree, and primitive hypotheses, shows the quotient-to-fraction-field route, and concludes that f is irreducible. — A primitive positive-degree polynomial satisfying the stated P and P² coefficient conditions is irreducible.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Separate Scientific inspection passed at original 1536 × 1024 pixels. The exact hypothesis labels are complete and legible: f in R[X] with P prime, leading coefficient outside P, lower coefficients in P below degree f, constant coefficient outside P², positive degree, and f.IsPrimitive. The large P region encloses precisely the lower coefficient cells, the leading cell is separate outside it, and the smaller P² region is nested away from the constant cell. The lower route reads R/P to FractionRing(R/P), notes that lower coefficients vanish, then reaches a pure monomial, generalized Eisenstein at q = X, and Irreducible f. The figure is a distinct flat Scientific diagram rather than a restyled concept image, and it makes no principal-ideal, maximal-ideal, UFD, field, monic, unit-leading-coefficient, or algorithm claim. — The prime-ideal coefficient pattern survives quotient and fraction-field passage as a pure monomial, while the constant term's exclusion from P² blocks a nontrivial factorization.

Complementary intuition

A mathematical landmark

Eisenstein's criterion is a fundamental and widely used test for polynomial irreducibility. Mathlib states it in a general prime-ideal form and derives it from a generalized quotient-and-fraction-field criterion, making the exact role of primitivity and the P² obstruction explicit.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem Polynomial.irreducible_of_eisenstein_criterion {R : Type*} [CommRing R] [IsDomain R] {f : R[X]} {P : Ideal R} (hP : P.IsPrime) (hfl : f.leadingCoeff ∉ P) (hfP : ∀ n : ℕ, ↑n < degree f → f.coeff n ∈ P) (hfd0 : 0 < degree f) (h0 : f.coeff 0 ∉ P ^ 2) (hu : f.IsPrimitive) : Irreducible f

Selected evidence declaration text

theorem Polynomial.irreducible_of_eisenstein_criterion {R : Type*} [CommRing R] [IsDomain R] {f : R[X]} {P : Ideal R} (hP : P.IsPrime) (hfl : f.leadingCoeff ∉ P) (hfP : ∀ n : ℕ, ↑n < degree f → f.coeff n ∈ P) (hfd0 : 0 < degree f) (h0 : f.coeff 0 ∉ P ^ 2) (hu : f.IsPrimitive) : Irreducible f

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's prime-ideal Eisenstein criterion for a polynomial over a commutative integral domain. The conclusion requires a prime ideal P, positive degree, primitivity, leading coefficient outside P, every coefficient strictly below the degree inside P, and constant coefficient outside P². It concludes irreducibility in R[X]; it is not an unrestricted prime-element or algorithmic criterion.

Proof Atlas did not originate Eisenstein's criterion or Mathlib's declaration.
The selected declaration does not assume or conclude that P is principal or maximal, R is a UFD or field, f is monic, or its leading coefficient is a unit.
The theorem does not provide a factorization or irreducibility algorithm, and it does not remove the primitivity or positive-degree hypotheses.
The generated explanation and visuals are not proof evidence.

Provenance

Origin and evidence stay separate

Existing declaration: Polynomial.irreducible_of_eisenstein_criterion in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.eisenstein-criterion.v001
Source handling: Verified upstream reference; no mirrored source package