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.
Read more about this topic: Algebraic Closure
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