Algebraic Closure - Examples

Examples

  • The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.
  • The algebraic closure of the field of rational numbers is the field of algebraic numbers.
  • There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(π).
  • For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order pn for each positive integer n (and is in fact the union of these copies).
  • See also Puiseux expansion.

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