Let $A$ be a finitely generated $K$-algebra, and let $\mathfrak p$ be a prime ideal of $A$ such that $A_{\mathfrak p}$ is an integral domain. Then have to show that $A_{\mathfrak p}$ is a localization of a finitely generated $K$-algebra which is a domain. (Here $K$ is a field.)
I don't know how to proceed. Please help me. Thanks.
Consider the map $A\to A_p$ and let $I$ be its kernel. Since $I$ is finitely generated, there exists an $s\in A-p$ such that $sI=0$. Then, the map $A_s\to A_p$ is injective and in particular, $A_s$ is a domain. $A_s$ is a finitely generated $K$-algebra, $(A_s)_p=A_p$ and so you are done.
$A=K[X_1,\dots,X_n]/I$, and $\mathfrak p=P/I$ with $P\subset K[X_1,\dots,X_n]$ a prime ideal containing $I$. We have that $A_{\mathfrak p}=K[X_1,\dots,X_n]_P/I_P$ is an integral domain. Then $I_P$ is prime, so $I_P=Q_P$ where $Q\subset K[X_1,\dots,X_n]$ is a prime with $Q\subseteq P$. Then $A_{\mathfrak p}=B_{\mathfrak q}$ where $B=K[X_1,\dots,X_n]/Q$ is a $K$-affine domain and $\mathfrak q=P/Q$.