The Jordan-Chevalley decomposition says that given a linear operator $L$, you can decompose it as $L = S + N$, where $S$ is diagonalizable and $N$ is nilpotent.
My textbook (Linear Algebra by Peterson) has a corollary of the Jordan-Chevalley that says that given $p(t) = (t-\lambda)^n$, the associated companion matrix $C_p$ is similar to a Jordan block (matrix with $\lambda$ on diagonal and $1$'s on superdiagonal).
So if I apply the JC decomposition to $C_p$, I get $C_p = \lambda I_n + (C_p - \lambda I_n) $. So I need to show that $C_p - \lambda I_n$ is similar to the matrix with $1$'s on the superdiagonal. I don't see how to do this without going into the Frobenius Canonical form and showing that the characteristic polynomial and minimal polynomial of $C_p$ are the same. (I.e. Using theorem that says if two operators have same minimal polynomial, then they are similar)
Is there an easy way to do this?