I've been reading through Humphreys Linear Algebraic Groups, and this question concerns the proof of Proposition 18.2 (page 117). I believe the essence of my confusion is the following claim:
$\textbf{Claim:}$ Let $X$ be an affine subvariety of $\operatorname{End}(V)$ for some finite dimensional $\mathsf{k}$-vector space $V$, and let $p \in \mathsf{K}[t]$ be monic of degree $\operatorname{dim}(V)$. Then the set
$$ W = \left\{ x \in X \mid c_x = p \right\}, $$
is closed in $X$, where $c_x$ denotes the characteristic polynomial of $x$.
Is this even true, and if so how would might I go about proving it?
Edit: I have an idea, does the following sound about right. The map $c : \operatorname{End}(V) \to \mathsf{k}[t]$ sending $x$ to $c_x$ is a polynomial in the coordinates of $\operatorname{End}(V)$, and so $W = c^{-1}(p) \cap X$ is closed?
Yes, this is true. First, we note that the characteristic polynomial has coefficients which are exactly polynomial functions in the entries of the matrix. Next, we note that asking $c_x=p$ is equivalent to enforcing the finite list of equalities given by equating coefficients. So the locus of matrices which satisfy this is exactly given as the variety associated to this ideal we just described, and is therefore closed.