An algebraically closed field with characteristic $p>0$

Question

I want to know about an algebraically closed field that is not of characteristic $0$.

I really don't know about infinite fields with characteristic $p$ so I will appreciate your comments.

Answer 1

Any field has an algebraic closure—so the short answer to this is: Just take the algebraic closure of any field of characteristic $p$.

For finite fields, it is possible to describe the algebraic closure fairly explicitly. Let $K=\mathbb{F}_q$, the finite field of order $q$. Then, for any $n\geq 1$, there is a unique (up to isomorphism) field extension $K\to K_n \cong \mathbb{F}_{q^n}$ of degree $n$, with $K_n \subset K_m$ if and only if $n | m$. $K_n$ can be described as the splitting field of any irreducible polynomial of degree $n$ with coefficients in $K$.

Clearly, the algebraic closure of $K$ is just $\overline{K} = \bigcup_n K_n $, which we could also write as $\varinjlim K_n$.