Jordan canonical form from rational canonical form

Jordan canonical form from rational canonical form: Let $V$ be a finite-dimensional vector space over a field $F$, and let $T:V\to V$ be linear. Assume the characteristic polynomial of $T$ splits over $F$ (for example, $F$ is algebraically closed). Then there exists a basis of $V$ for which the matrix of $T$ is in Jordan canonical form.

This follows by refining the invariant-factor decomposition in rational canonical form from the structure theorem into primary factors, yielding the Jordan canonical form theorem; the Jordan blocks are organized by the roots of the characteristic polynomial.