Algebraic closure

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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.

The definition of algebraically closed has several other equivalent forms:

  1. For every and every polynomial f(x), there is a such that .
  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. Every algebraic variety defined over F has a point.
  7. F is not finite and the automorphism group of F has cardinality strictly greater than the cardinality of F.

The complex numbers are algebraically closed by the Fundamental Theorem of Algebra, however the rational numbers are not since x2+1 does not have a root. This same polynomial shows the real numbers are also not algebraically closed, yielding as a corollary (by the equivalence) that has cardinality the continuum.