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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?

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    If $\mathfrak{p}$ is maximal then the residue field is $A/\mathfrak{p}$ which is $k$ by the (weak) Nullstellensatz. Have you tried any examples, e.g. $A = k[x, y], \mathfrak{p} = (x)$?2012-08-06

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