Is there an algebraic characterization of when a 2 x 2 matrix is diagonalizable?

Question

All matrices are over the complex numbers. There is, of course, an algebraic characterization of when a 2x2 matrix $$ \left( \begin{array}{cc} a & b\\ c & d\\ \end{array} \right) $$ is not invertible: $ad - bc = 0$. Is there a similar one for when that matrix is diagonalizable? more specifically I mean, is there a polynomial $p \in \mathbb{C}[a,b,c,d]$ such that $p(A) = 0$ iff $A$ is diagonalizable (or $p(A) = 0$ iff $A$ is not diagonalizable)?

Edit: The $p$ should be universal, and apply to every matrix - that is, I'd like to find out whether there exists a $p \in \mathbb{C}[a,b,c,d]$ such that $\{A \mid A \text{ is not diagonalizable}\} = \{A \mid p(A) = 0\}$ or $\{A \mid p(A) \neq 0\}$.

Answer 1

For an algebraically closed field like $\mathbb{C}$ a matrix $A$ is diagonalizable if and only if $\gcd(p,p^{'})=1$, where $p$ is the minimal polynomial of $A$ and $p^{'}$ is its derivative.

Answer 2

For $2\times2$ matrices, it is not so hard. If it is not diagonalizable, then the characteristic polynomial must have a double root. This is tested by seeing if $\text{trace}(A)^2 - 4 \det(A) = 0$ (that is, the discriminant of the quadratic characteristic polynomial is zero).

Now if the characteristic polynomial has a double root, then it will be non-diagonalizable if and only if either off diagonal entry is non-zero.

Since one of the tests is that something is zero, and the other test is that something is not zero, I don't see how to get this in the form $p(A) = 0$.