Let $k$ be an algebraically closed field of characteristic $p$ and $A=k[x_1,\cdots,x_n]$ the polynomial ring over $k$ in $n$ variables.
Given a prime ideal $\mathfrak{p}$ in $A$, denote by $A_\mathfrak{p}$ the localization in $\mathfrak{p}$ and also denote by $A(\mathfrak{p})$ the residue field of the local ring $A_\mathfrak{p}$, i.e. $A(\mathfrak{p})=A_\mathfrak{p}/\mathfrak{p}A_\mathfrak{p}$.
Is there a way to describe explicitly $A(\mathfrak{p})$? What about the case where $\mathfrak p$ is maximal?