Algebraic closure

A field F is algebraically closed if there is a solution to every polynomial over F in one variable. Every field is contained in an algebraically closed field and there is a minimal one, called the algebraic closure of F. This general fact uses Zorn's Lemma, which is equivalent to the Axiom of Choice (AC). In specific cases AC may sometimes be avoided.

The definition of algebraically closed has several other equivalent forms:

1. For every $a\in F$ and every polynomial f(x), there is a $b\in F$ such that $f(b)=a$.
2. The only irreducible polynomials in F[x] have degree 1.
3. F has no finite extensions other than itself.
4. F is a maximal field of its transcendence degree.
5. F is the union of all its finitely-generated localized subrings.
6. F is not finite and the automorphism group of F has cardinality strictly greater than the cardinality of F.

The complex numbers $\mathbb {C}$ are algebraically closed by the Fundamental Theorem of Algebra, however the rational numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}} are not since x²+1 does not have a root. This same polynomial shows the real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} are also not algebraically closed, yielding as a corollary (by the equivalence) that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Aut}(\mathbb{R})} has cardinality the continuum.

The complex numbers are the algebraic closure of both the real numbers and the rational numbers.