Fake Proofs in Mathematics

Consider the $F$-linear transformation $$T_\alpha:E\longrightarrow E, \qquad T_\alpha(v)=\alpha v.$$ Let $m_\alpha(x)$ be the minimal polynomial of $\alpha$ over $F$. This is also the minimal polynomial of the linear transformation $T_\alpha$.

By the Cayley-Hamilton theorem, $m_\alpha(x)$ divides the characteristic polynomial $\chi_{T_\alpha}(x)$. Since $$\deg\chi_{T_\alpha}=[E:F]=p,$$ it follows that $$\deg m_\alpha\mid p.$$ Because $\alpha\notin F$, we have $\deg m_\alpha>1$. Since $p$ is prime, this forces $\deg m_\alpha=p$. Therefore $$[F(\alpha):F]=p=[E:F],$$ and hence $E=F(\alpha)$.