Let $R$ be an algebra of finite type over a field $k$. I want to show that $R$ is Jacobson, i.e. that any prime ideal $\mathfrak{p}$ in $R$ is an intersection of maximal ideals. I am not getting very far however.
I know that the Jacobson radical of an algebra of finite type over a field $k$ is the nilradical, or in other words, the intersection of all maximal ideals is the same as the intersection of all prime ideals. So my first thought was to try to show that any prime ideal $\mathfrak{p}$ is the intersection of all maximal ideals which contain $\mathfrak{p}$. However if this is the case, I am not sure how to show the intersection of all such maximal ideals is actually $\subseteq \mathfrak{p}$.
Any help would be appreciated with this, thanks.