isConjRoot_algHom_iff — Mathlib

English

@[simp] If s is an R-algebra isomorphism, then IsConjRoot R x (s x).

Русский

@[simp] если s — R-алгебра изоморфизм, то IsConjRoot R x (s x).

LaTeX

$$$IsConjRoot(R, x, s(x)) = \minpoly(R, x) = \minpoly(R, s(x))$$$

Lean4

/-- If `y` is a conjugate root of `x` in some division ring and `f` is a `R`-algebra homomorphism, then
`f y` is a conjugate root of `f x`.
-/
theorem isConjRoot_algHom_iff {A} [DivisionRing A] [Algebra R A] [Nontrivial B] {x y : A} (f : A →ₐ[R] B) :
    IsConjRoot R (f x) (f y) ↔ IsConjRoot R x y :=
  isConjRoot_algHom_iff_of_injective f.injective

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