Split characteristic polynomials decompose into generalised eigenspaces of the algebraic multiplicities

Fact [L1] makes ${\mu }_T$ split over $F$, so its distinct monic irreducible factors are the $x-\lambda$ with $\lambda$ an eigenvalue and $G_{\lambda }=\ker (T-\lambda I{)}^{e_{\lambda }}$. Fact [L2] gives the displayed invariant direct sum, and [L3] applied to the same factorisation gives that each $T{\mathrm{\mid }}_{G_{\lambda }}$ has minimal polynomial exactly $(x-\lambda {)}^{e_{\lambda }}$.

Fix an eigenvalue $\lambda$. Since $\lambda$ is an eigenvalue, $\ker (T-\lambda I)\mathrm{\ne }0$, and $\ker (T-\lambda I)\subseteq G_{\lambda }$ because $e_{\lambda }\ge 1$; hence $G_{\lambda }$ is nonzero and [L5] applies to it. On $G_{\lambda }$, $N_{\lambda }=T{\mathrm{\mid }}_{G_{\lambda }}-\lambda I$ has minimal polynomial $x^{e_{\lambda }}$ by step 1.1, so [L5] makes $N_{\lambda }$ nilpotent of index $e_{\lambda }$ and gives ${\chi }_{N_{\lambda }}=x^{\dim G_{\lambda }}$. Translating the scalar variable yields ${\chi }_{T{\mathrm{\mid }}_{G_{\lambda }}}=(x-\lambda {)}^{\dim G_{\lambda }}$.

Repeated use of [L4] on the invariant direct sum multiplies these restricted characteristic polynomials to obtain ${\chi }_T={\prod }_{\lambda }(x-\lambda {)}^{\dim G_{\lambda }}$, so the exponent is exactly the algebraic multiplicity. The empty product gives the zero-space case.