finite field cannot be algebraically closed

Theorem.

A finite field cannot be algebraically closed.

Proof.

The proof proceeds by the method of contradiction. Assume that a field$F$ is both finite and algebraically closed. Consider the polynomial$p(x)=x^2-x$ as a function from $F$ to $F$. There are two elementswhich any field (in particular, $F$) must have — the additive identity$0$ and the multiplicative identity $1$. The polynomial $p$ maps both ofthese elements to $0$. Since $F$ is finite and the function $p:F\to F$ is not one-to-one, the function cannot map onto $F$ either, sothere must exist an element $a$ of $F$ such that $x^2-x\ne a$ forall $x\in F$. In other words, the polynomial $x^2-x-a$ has no rootin $F$, so $F$ could not be algebraically closed.∎