Exercise 8.C.16

Proof. We have

Taking the adjoint of it side yields

and we see that the minimal polynomial of $T^{\text{∗}}$ is

To see this, suppose by contradiction this is not the minimal polynomial of $T^{\text{∗}}$. Let $p$ denote the minimal polynomial $T$. Then $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p<m$. Because $p(T^{\text{∗}})=0$, taking the adjoing of each side as we did above shows that $\overline{p}(T)=0$, where $\overline{p}$ equals $p$ with conjugated coefficients. But $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->\overline{p}<m$, which is a contradiction because the minimal polynomial of $T$ has degree $m$. □