Identifying $M_n(k)$ with $k^{n^2}$ with $k$ algebraically closed, I am asked to show that the subset of diagonalizable matrices, $D_n$ is constructible. Constructible is defined as being the finite union of locally closed sets.
I understand that the set of matrices with characterstic polynomials having multiple roots is closed, and its complement is an open set (hence constructible) contained in $D_n$. But I am not sure where to go from here.
Observe that $D_n \subseteq k^{n^2}$ is a definable set in the field language. That is, there is a formula of first-order logic, $\phi(\overline{x_{ij}})$, such that $k\models \phi(\overline{a_{ij}})$ if and only if $(a_{ij})\in D_n$.
We just need to write down $\exists (y_{ij}),\exists (y_{ij}'), \exists (z_i), ((y_{ij})(y_{ij}') = I \land (y_{ij})(x_{ij})(y_{ij}') = Z)$, where $Z$ is the diagonal matrix with the values $z_i$ along the diagonal. I've abbreviated by not writing out the polynomials which define matrix multiplication.
Now by quantifier elimination for algebraically closed fields, the formula $\phi$ is equivalent to a quantifier-free formula $\psi$. A quantifier-free formula in the language of fields is just a Boolean combination (i.e. $\land$, $\lor$, $\lnot$) of polynomial equations, so it picks out a Boolean combination (i.e. $\cap$, $\cup$, $^c$) of Zariski-closed sets, i.e. a constructible set.
If you don't like the language of model theory I used above, you can translate it to the language of algebraic geometry using Chavalley's theorem (which is just another way of saying quantifier elimination for algebraically closed fields): the image of a constructible set under a morphism of varieties is a constructible set.
The subset of $k^{n^2+n^2+n^2 + n}$ where the first $n^2$ coordinates are a matrix $A$, the second $n^2$ coordinates are a matrix $B$, the third $n^2$ coordinates are a matrix $C$, the last $n$ coordinates are the diagonal entries of a diagonal matrix $D$, such that $C = B^{-1}$ and $BAC = D$, is Zariski-closed, so its image under projection to the first $n^2$ coordinates ($D_n$) is constructible.