Does there exists a finite field such is algebraically closed?

I tried of course the most obvious example $\displaystyle \mathbb{Z}_2$ but it is not algebraically closed for $\displaystyle x^2+x+1\in F[\mathbb{Z}_2]$ has no zero.

This problem seems strongly connected with number theory because it invloves prime numbers because all finite fields have order $\displaystyle p^n$.