Theorems · Definition · commutative algebra

IsAdjoinRoot.algEquiv

{R : Type u} →
  {S : Type v} →
    [inst : CommRing R] →
      [inst_1 : Ring S] →
        {f : Polynomial R} →
          [inst_2 : Algebra R S] →
            IsAdjoinRoot S f →
              {T : Type u_1} → [inst_3 : Ring T] → [inst_4 : Algebra R T] → IsAdjoinRoot T f → S ≃ₐ[R] T

Adjoining a root gives a unique algebra up to isomorphism. This is the converse of IsAdjoinRoot.ofAlgEquiv: this turns an IsAdjoinRoot into an AlgEquiv, and IsAdjoinRoot.ofAlgEquiv turns an AlgEquiv into an IsAdjoinRoot.

Defined in

Mathlib.RingTheory.IsAdjoinRoot

Cited by

12 results in Mathlib

Foundations

Depth 109 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingRingAlgebraRingAlgebra