In Miles Reid, undergraduate commutative algebra, I read the following:
"Suppose that $k$ is an algebraically closed field and that $A=k[x_1,...,x_n]$ is a finitely generated $k$-algebra of form $A=k[X_1,...,X_n]/J$ where $J$ is an ideal of $k[X_1,...,X_n]$. (Here I'm using the notation to mean that $x_i=X_i$ mod $J$). Then every maximal ideal of $A$ is of form $(x_1-a_1,...,x_n-a_n)$ for some point $(a_1,...,a_n) \in V(J)$"
Have I understood it right if I say $x_i \in k[X_1,...,X_n]/J$?
Can someone explain what he means by "Here I'm using the notation....".
$k$ is algebraically closed, what does this imply? Why do we need this?