Exercise 8.C.9

Proof. Since

multiplying both sides by $T^{-5}$ we get

Hence $4z^5+5z^4-6z^3-7z^2+2z+1$ is a polynomial multiple of the minimal polynomial of $T^{-1}$ (by 8.46). As it turns out, this is actually the minimal polynomial of $T^{-1}$ (with the coefficients multiplied by $4$). To see this, suppose by contradiction that it is not. Hence, the minimal polynomial of $T$ has degree at most $4$. This means that

for some $a_0,a_1,a_2,a_3,a_4\in 𝔽$, not all equal $0$. Multiplying both sides of the equation above by $T^4$, we get

which is a contradiction, because the minimal polynomial of $T$ has degree $5$. □