AlgebraicClosure — Mathlib

English

The canonical algebraic closure of a field k is defined as the quotient of the polynomial ring by the maximal ideal: AlgebraicClosure(k) = MvPolynomial(Vars(k), k) / maxIdeal k.

Русский

Каноническое алгебраическое замыкание поля k определяется как тождество MvPolynomial(Vars(k), k) / maxIdeal k.

LaTeX

$$$\text{AlgebraicClosure}(k) = \operatorname{MvPolynomial}(\mathrm{Vars}(k), k) / \maxIdeal k.$$$

Lean4

/-- The canonical algebraic closure of a field, the direct limit of adding roots to the field for
each polynomial over the field. -/
@[stacks 09GT]
def AlgebraicClosure : Type u :=
  MvPolynomial (Vars k) k ⧸ maxIdeal k

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