# How to show that $D_n$ is closed in $GL_n$ with respect to Zariski topology?

Let $GL_n$ be the group of all invertible matrices of order $n$ and $D_n$ the subgroup of $GL_n$ consisting of all invertible diagonal matrices of order $n$. How to show that $D_n$ is closed in $GL_n$ with respect to Zariski topology? I think that we have to show that $D_n$ is the set of zeros of some set of polynomials. What are these polynomials? Thank you very much.

• Hint: the function which returns the $(i,j)$-th entry of a matrix is a polynomial function on $\mathrm{GL}_n$. – mdp Sep 25 '13 at 9:46
• Do you really mean for your diagonal matrices to have order $n$? – Tobias Kildetoft Sep 25 '13 at 9:48
• @MattPressland, thank you very much. I think that $D_n=\{g=(x_{ij}) : f_{ij}(g) = x_{ij}=0, i\neq j, i, j \in \{1, \ldots, n\}\}$. – LJR Sep 25 '13 at 9:52
• @TobiasKildetoft, thank you very much. I mean that the diagonal matrices are $n$ by $n$ matrices. – LJR Sep 25 '13 at 9:55

Hint: the function which returns the $(i,j)$-th entry of a matrix is a polynomial function on $\mathrm{GL}_n$.