Let $T:V \rightarrow V$ be an linear operator. If $T = T_{1} \bigoplus T_{2}$ then $p_T(x)=p_{T_{1}}(x)\cdot p_{T_{2}}(x)$ where $p_T$ is the characteristic polynomial of T.
My attempt: By Cayley-Hamilton theorem, $p_T(T) = p_{T}(T_{1} \bigoplus T_{2}) = p_{T}(T_{1}) \bigoplus p_{T}(T_{2}) = 0$ then $p_{T}(T_1) = 0$ and $p_T(T_2) = 0$. So, we have that the minimal polynomial $m_{T_1}$ of $T_1$and $m_{T_2}$ of $T_2$ divides $p_T$, then $$p_T = \operatorname{lcm}(m_{T_1},m_{T_2}).$$ But how can I come up with $p_{T} = p_{T_1}p_{T_2}$? Does I need to use the fact that $\operatorname{dim}(T) = \operatorname{dim}(T_1)+\operatorname{dim}(T_2)$?
Any help is appreciated!