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# algebraic closure of a finite field

Fix a prime $p$ in $\mathbb{Z}$. Then the Galois fields $GF(p^{e})$ denotes the finite field of order $p^{e}$, $e\geq 1$. This can be concretely constructed as the splitting field of the polynomials $x^{{p^{e}}}-x$ over $\mathbb{Z}_{p}$. In so doing we have $GF(p^{{e}})\subseteq GF(p^{{f}})$ whenever $e|f$. In particular, we have an infinite chain:

$GF(p^{{1!}})\subseteq GF(p^{{2!}})\subseteq GF(p^{{3!}})\subseteq\cdots% \subseteq GF(p^{{n!}})\subseteq\cdots.$ |

So we define $\displaystyle GF(p^{{\infty}})=\bigcup_{{n=1}}^{\infty}GF(p^{{n!}})$.

###### Theorem 1.

$GF(p^{{\infty}})$ is an algebraically closed field of characteristic $p$. Furthermore, $GF(p^{e})$ is a contained in $GF(p^{{\infty}})$ for all $e\geq 1$. Finally, $GF(p^{\infty})$ is the algebraic closure of $GF(p^{e})$ for any $e\geq 1$.

###### Proof.

Given elements $x,y\in GF(p^{\infty})$ then there exists some $n$ such that $x,y\in GF(p^{{n!}})$. So $x+y$ and $xy$ are contained in $GF(p^{{n!}})$ and also in $GF(p^{\infty})$. The properties of a field are thus inherited and we have that $GF(p^{\infty})$ is a field. Furthermore, for any $e\geq 1$, $GF(p^{e})$ is contained in $GF(p^{{e!}})$ as $e|e!$, and so $GF(p^{e})$ is contained in $GF(p^{\infty})$.

Now given $p(x)$ a polynomial over $GF(p^{\infty})$ then there exists some $n$ such that $p(x)$ is a polynomial over $GF(p^{{n!}})$. As the splitting field of $p(x)$ is a finite extension of $GF(p^{{n!}})$, so it is a finite field $GF(p^{{e}})$ for some $e$, and hence contained in $GF(p^{\infty})$. Therefore $GF(p^{\infty})$ is algebraically closed. ∎

We say $GF(p^{\infty})$ 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.

## Mathematics Subject Classification

12F05*no label found*

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