# mathlibdocumentation

ring_theory.polynomial.tower

# Algebra towers for polynomial #

This file proves some basic results about the algebra tower structure for the type polynomial R.

This structure itself is provided elsewhere as polynomial.is_scalar_tower

theorem is_scalar_tower.aeval_apply (R : Type u) (S : Type v) (A : Type w) [semiring A] [ S] [ A] [ A] [ A] (x : A) (p : polynomial R) :
p = (polynomial.map S) p)
theorem is_scalar_tower.algebra_map_aeval (R : Type u) (A : Type w) (B : Type u₁) [semiring B] [ A] [ B] [ B] [ B] (x : A) (p : polynomial R) :
B) ( p) = (polynomial.aeval ( B) x)) p
theorem is_scalar_tower.aeval_eq_zero_of_aeval_algebra_map_eq_zero (R : Type u) (A : Type w) (B : Type u₁) [semiring B] [ A] [ B] [ B] [ B] {x : A} {p : polynomial R} (h : function.injective B)) (hp : (polynomial.aeval ( B) x)) p = 0) :
p = 0
theorem is_scalar_tower.aeval_eq_zero_of_aeval_algebra_map_eq_zero_field {R : Type u_1} {A : Type u_2} {B : Type u_3} [field A] [nontrivial B] [ A] [ B] [ B] [ B] {x : A} {p : polynomial R} (h : (polynomial.aeval ( B) x)) p = 0) :
p = 0
@[simp]
theorem subalgebra.aeval_coe (R : Type u) {A : Type w} [ A] {S : A} {x : S} {p : polynomial R} :
p = ( p)