Proof: If $F^3 = F$ then F is diagonalisable

Question

let $V$ be a $\mathbb{R}$-vectorspace with $dim V < \infty$ and $F$ an endomorphism of V with $F^3 = F$.

Show: F is diagonalisable.

• $F^3 = F$ is equivalent to $F^3 - F = 0$.

• Now I know that $F$ is diagonalisable if the minimal-polynomial has linear-factors and every eigen-value is only a singular-null of the polynomial.

• Now I should be able to conclude from $F^3 - F = 0$ that the minimal-polynomial... ?

Here I stuck :(

Answer 1

Let $p(x)=x^3-x=x(x-1)(x+1)$ and note that the minimal polynomial of $F$ divides $p(x)$. But $p(x)$ is a product of linear factors so the minimal polynomial itself must be a product of linear factors.