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 .
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 .
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.
The algebraic closure of a finite field is countable.
By construction the algebraic closure is a countable union of finite sets so it is countable. ∎
- 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|
|Date of creation||2013-03-22 16:40:51|
|Last modified on||2013-03-22 16:40:51|
|Last modified by||Algeboy (12884)|