Minimal polynomial?

Question

Let $K$ a field, $p,n\in \mathbb{N}$, $B\in \mathcal{M}_p({K})$ and let denote $S_B=\{X\in \mathcal{M}_p(K) \ \mid \ X^n=B\}$.

If $X\in S_B$, I have to prove that $\mu_X$ (the minimal polynomial of $X$) divides $\mu_B(\xi^n)$.

If $X\in S_B$ then we can deduce that $X^n-B=0$ but after I do not know how to link this with $\mu_X$ or $\mu_B$...

Thanks in advance !

Answer 1

Hint: It suffices to note that if $X \in S_B$, then $\mu_B(X^n) = 0$