\item WLOG let the minimal polynomial of \(\alpha\) be monic, ie \(\alpha^n + \sum a_i\alpha^i = 0\). Let \(\{1, \alpha, \dots, \alpha^{n-1}\}\) be a basis for \(K(\alpha)\). have that \(T(\alpha^i) = \alpha^{i+1}\) for \(i \in \{0,1,2,3,...,n-2\}\). Now \(T(\alpha^{n-1}) = \alpha^n = -\sum a_i\alpha^i\). So we have that
    			\begin{align*}
    				T \to \begin{bmatrix}
    				0&0&0&...&0&-a_0\\
    				1&0&0&...&0&-a_1\\
    				0&1&0&...&0&-a_2\\
    				0&0&1&...&0&-a_3\\
    				\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
    				0&0&0&...&1&-a_{n-2}\\
    				0&0&0&...&0&-a_{n-1}\\
    				\end{bmatrix}
    			\end{align*}