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Let $R=\mathbb{Q}[x_{i,j}\,:\, 1\leq i,j\leq n]$. Let $M$ be the $n\times n$ matrix $(x_{i.j})$. Let $\chi(M)$ be the characteristic polynomial of $M$. Finally, let $I$ be the ideal of $R$ generated by the non-constant coefficients of $\chi(M)$.

It should be well known that $I$ is a prime ideal of $R$. Is there a good reference for this fact?

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    $\begingroup$ You are asking for a reference proving that the nilpotent cone is integral (irreducible and reduced). $\endgroup$
    – Jason Starr
    Commented May 21, 2014 at 18:40
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    $\begingroup$ http://en.wikipedia.org/wiki/Nilpotent_cone $\endgroup$
    – Jason Starr
    Commented May 21, 2014 at 18:41
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    $\begingroup$ The better theorem (since it's obviously connected, being a cone) is that it's normal, due to Kostant. $\endgroup$
    – Allen Knutson
    Commented May 24, 2014 at 18:15

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