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Let $F$ be an arbitrary field of characteristic $0$, $K$ its algebraic closure. Define $M=\{ (x,y)\in M_n(F)×M_n(F) \mid [x,y]=0\}$ and let $N$ be the Zariski closure of $M$ in $K^{2n^2}$.

How can one show that $N$ contains the set $\{(axa^{-1},aya^{-1}) \mid (x,y)\in N, a\in \mathrm{GL}(n,K)\}$?

Thank you.

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    Show that $\mathrm{GL}_n(F)$ is dense in $\mathrm{GL}_n(K) \subset K^{n^2+1}$ for the Zariski topology.2011-05-06

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