Let B be a nilpotent linear operator with index p. Is operator $ N^2 + \alpha*N $ nilpotent?

Question

Let B be a nilpotent linear operator with index p. Is operator $ N^2 + \alpha*N $ nilpotent? ( $ \alpha \in \mathbb{R}$) I think that when $ \alpha = 0$ then the index is $p/2$ for even p, but when p is odd I'm not sure. It is obviously nilpotent but the index is what? It can't be $p/2$ because that is not a natural number. When $ \alpha \neq 0$ then the index is p? I used the binomial theorem.

Answer 1

Since the minimal polynomial of $N$ is $t^p$, you should find the first $k$ such that $$ t^p\ |\ (t^2+\alpha t)^k. $$ If $\alpha = 0$ and $p$ is odd, $k$ is $\lceil \frac{p}2\rceil=\frac{p+1}2$. If $\alpha\ne 0$, then $k$ should be $p$.