algebraic closure of a finite field
Fix a prime in . Then the Galois fields denotes the finite field of order , . This can be concretely constructed as the splitting field of the polynomials over . In so doing we have whenever . In particular, we have an infinite chain:
So we define .
Theorem 1.
is an algebraically closed field of characteristic . Furthermore, is a contained in for all . Finally, is the algebraic closure of for any .
Proof.
Given elements then there exists some such that . So and are contained in and also in . The properties of a field are thus inherited and we have that is a field. Furthermore, for any , is contained in as , and so is contained in .
Now given a polynomial over then there exists some such that is a polynomial over . As the splitting field of is a finite extension of , so it is a finite field for some , and hence contained in . Therefore is algebraically closed. ∎
We say 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 countable.
Proof.
By construction the algebraic closure is a countable union of finite sets so it is countable. ∎
References
- 1 McDonald, Bernard R., Finite rings with identity, 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 |