English
A ring is algebraic over A iff it is algebraic over the fraction field of A, preserved under comaps.
Русский
Кольцо алгебраично над A тогда и только тогда над полем дробей A, сохранением подобразования.
LaTeX
$$$\text{Algebra.IsAlgebraic}_A \iff \text{Algebra.IsAlgebraic}_K$ через комап.$$
Lean4
/-- An element of a ring is algebraic over the ring `A` iff it is algebraic
over the field of fractions of `A`.
-/
theorem isAlgebraic_iff [Algebra A C] [Algebra K C] [IsScalarTower A K C] {x : C} : IsAlgebraic A x ↔ IsAlgebraic K x :=
by
constructor <;> rintro ⟨p, hp, px⟩
· refine ⟨p.map (algebraMap A K), fun h => hp (Polynomial.ext fun i => ?_), ?_⟩
· have : algebraMap A K (p.coeff i) = 0 := _root_.trans (Polynomial.coeff_map _ _).symm (by simp [h])
exact to_map_eq_zero_iff.mp this
· exact (Polynomial.aeval_map_algebraMap K _ _).trans px
·
exact
⟨integerNormalization _ p, mt integerNormalization_eq_zero_iff.mp hp, integerNormalization_aeval_eq_zero _ p px⟩
