Prove that the field $\mathbb{F}_{p}$ is not algebraically closed where $p$ is prime

Question

How would you go about proving that the field $\mathbb{F}_{p}$ is not algebraically closed where $p$ is prime?

Any help would be great!

Answer 1

Let the polynomial $f\in \Bbb F_p$ be given as $$ f(x) = x(x-1)(x-2)\cdots (x-p+1) $$ Then $f(a) = 0$ for any $a \in \Bbb F_p$. Thus $g(x) = f(x) + 1$ has no roots in $\Bbb F_p$, and therefore $\Bbb F_p$ has a non-trivial algebraic extension obtained by adjoining all the roots of $g$.

The same proof (with minor changes to the definition of $f$) goes for any finite field, not just the ones of prime order. This shows that a finite field can never be algebraically closed.