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data.polynomial.integral_normalization

Theory of monic polynomials #

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

noncomputable def polynomial.integral_normalization {R : Type u} [semiring R] (f : R[X]) :

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

Moreover, integral_normalization 0 = 0.

Equations
theorem polynomial.integral_normalization_coeff {R : Type u} [semiring R] {f : R[X]} {i : } :
theorem polynomial.integral_normalization_eval₂_eq_zero {R : Type u} {S : Type v} [comm_ring R] [is_domain R] [comm_semiring S] {p : R[X]} (f : R →+* S) {z : S} (hz : polynomial.eval₂ f z p = 0) (inj : ∀ (x : R), f x = 0x = 0) :
theorem polynomial.integral_normalization_aeval_eq_zero {R : Type u} {S : Type v} [comm_ring R] [is_domain R] [comm_semiring S] [algebra R S] {f : R[X]} {z : S} (hz : (polynomial.aeval z) f = 0) (inj : ∀ (x : R), (algebra_map R S) x = 0x = 0) :