algebraicClosure — Mathlib

English

The collection of elements algebraic over the base ring forms a subalgebra of the ambient algebra.

Русский

Сбор алгебраических над оснR кольцом элементов образует подалгебру над R.

LaTeX

$$def algebraicClosure (R S) := { s ∈ S | IsAlgebraic R s }$$

Lean4

/-- If `R` is a domain and `S` is an arbitrary `R`-algebra, then the elements of `S`
that are algebraic over `R` form a subalgebra. -/
def algebraicClosure [IsDomain R] : Subalgebra R S
    where
  carrier := {s | IsAlgebraic R s}
  mul_mem' ha hb := ha.mul hb
  add_mem' ha hb := ha.add hb
  algebraMap_mem' := isAlgebraic_algebraMap

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