TITLE: Algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$
QUESTION [8 upvotes]: Is there a concrete description of the algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$?

REPLY [10 votes]: First notice that the required algebraic closure must contain an algebraic closure $\overline { \mathbb F_p}$ of $\mathbb F_p$, so that we must actually find an algebraic closure of $\overline { \mathbb F_p}(t)$.  
For an algebraically closed field $k$ of characteristic zero (like $\mathbb C$) an algebraic closure of $k(T)$ is the algebraic closure of  $k(T)$ in the algebraically closed field of Puiseux series $$k((T^{\frac{1}{\infty}}))=\bigcup_{n\geq 1} k((T^{\frac{1} 
{n}}))$$  
This is no longer true if $char. k\gt 0$, as Chevalley showed by exhibiting as a counterexample the polynomial $Y^p-Y-T^{-1}\in k(T)[Y]$, which has  no root in $k((T^{\frac{1}{\infty}}))$. 
Fortunately   a modification of the notion of Puiseux series gives an analogue in characteristic $p$ of the above algebraic closedness result and that should, in principle, solve your problem.
 You could consult Kedlaya's paper for details.