algebraic closure of a finite field

Fix a prime p in . Then the Galois fields GF(pe) denotes the finite field of order pe, e1. This can be concretely constructed as the splitting fieldMathworldPlanetmath of the polynomialsPlanetmathPlanetmath xpe-x over p. In so doing we have GF(pe)GF(pf) whenever e|f. In particular, we have an infinite chain:


So we define GF(p)=n=1GF(pn!).

Theorem 1.

GF(p) is an algebraically closed field of characteristicPlanetmathPlanetmath p. Furthermore, GF(pe) is a contained in GF(p) for all e1. Finally, GF(p) is the algebraic closureMathworldPlanetmath of GF(pe) for any e1.


Given elements x,yGF(p) then there exists some n such that x,yGF(pn!). So x+y and xy are contained in GF(pn!) and also in GF(p). The properties of a field are thus inherited and we have that GF(p) is a field. Furthermore, for any e1, GF(pe) is contained in GF(pe!) as e|e!, and so GF(pe) is contained in GF(p).

Now given p(x) a polynomial over GF(p) then there exists some n such that p(x) is a polynomial over GF(pn!). As the splitting field of p(x) is a finite extensionMathworldPlanetmath of GF(pn!), so it is a finite field GF(pe) for some e, and hence contained in GF(p). Therefore GF(p) is algebraically closed. ∎

We say GF(p) is the algebraic closure indicating that up to field isomorphisms, there is only one algebraic closure of a field. The actual objects and constructions may vary.

Corollary 2.

The algebraic closure of a finite field is countableMathworldPlanetmath.


By construction the algebraic closure is a countable union of finite setsMathworldPlanetmath so it is countable. ∎


  • 1 McDonald, Bernard R., Finite rings with identityPlanetmathPlanetmathPlanetmath, Pure and Applied Mathematics, Vol. 28, Marcel Dekker Inc., New York, 1974, p. 48.
Title algebraic closure of a finite field
Canonical name AlgebraicClosureOfAFiniteField
Date of creation 2013-03-22 16:40:51
Last modified on 2013-03-22 16:40:51
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 5
Author Algeboy (12884)
Entry type Derivation
Classification msc 12F05
Related topic FiniteField
Related topic FiniteFieldCannotBeAlgebraicallyClosed