Documentation

Mathlib.RingTheory.Polynomial.IntegralNormalization

Theory of monic polynomials #

We define integralNormalization, which relate arbitrary polynomials to monic ones.

noncomputable def Polynomial.integralNormalization {R : Type u} [Semiring R] (f : Polynomial R) :

If f : R[X] is a nonzero polynomial with root z, integralNormalization f is a monic polynomial with root leadingCoeff f * z.

Moreover, integralNormalization 0 = 0.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    theorem Polynomial.integralNormalization_eval₂_eq_zero {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommSemiring S] {p : Polynomial R} (f : R →+* S) {z : S} (hz : Polynomial.eval₂ f z p = 0) (inj : ∀ (x : R), f x = 0x = 0) :
    theorem Polynomial.integralNormalization_aeval_eq_zero {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommSemiring S] [Algebra R S] {f : Polynomial R} {z : S} (hz : ↑(Polynomial.aeval z) f = 0) (inj : ∀ (x : R), ↑(algebraMap R S) x = 0x = 0) :