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Mathlib.RingTheory.EisensteinCriterion

Eisenstein's criterion #

A proof of a slight generalisation of Eisenstein's criterion for the irreducibility of a polynomial over an integral domain.

theorem Polynomial.irreducible_of_eisenstein_criterion {R : Type u_1} [CommRing R] [IsDomain R] {f : Polynomial R} {P : Ideal R} (hP : P.IsPrime) (hfl : f.leadingCoeffP) (hfP : ∀ (n : ), n < f.degreef.coeff n P) (hfd0 : 0 < f.degree) (h0 : f.coeff 0P ^ 2) (hu : f.IsPrimitive) :

If f is a non constant polynomial with coefficients in R, and P is a prime ideal in R, then if every coefficient in R except the leading coefficient is in P, and the trailing coefficient is not in P^2 and no non units in R divide f, then f is irreducible.