Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms

Let $V$ be a finite-dimensional vector space over a field $F$, and let $T:V\to V$ be linear. The endomorphism $T$ is semisimple when its minimal polynomial is separable in the sense of Repeated roots in extension fields and separable polynomials.

Equivalently, choose any basis of $V$, regard the matrix of $T$ as a matrix over an algebraic closure $\bar{F}\mathrm{/}F$ from An algebraic closure of a field, and let it act on ${\bar{F}}^n$. That endomorphism is diagonalisable over $\bar{F}$. This is basis-independent because the minimal polynomial is unchanged by field extension, and an endomorphism is diagonalisable exactly when its minimal polynomial splits with distinct roots.

The endomorphism $T$ is nilpotent when $T^m=0$ for some positive integer, equivalently when it is nilpotent in the sense of Nilpotent endomorphisms and their nilpotency index.