1. Let K be a field extension of F of prime degree p and let $\displaystyle \alpha$ be an element of K which is not in F. Prove that K = F($\displaystyle \alpha$)
im not sure how to solve this
We have $\displaystyle p=[K:F]=[K:F(\alpha)]\times[F(\alpha):F]$. Thus $\displaystyle [F(\alpha):F] = p$ or 1. Since $\displaystyle F(\alpha)\supsetneq F$, we can’t have $\displaystyle [F(\alpha):F] = 1$; therefore $\displaystyle [F(\alpha):F] = p$ $\displaystyle \Rightarrow$ $\displaystyle [K:F(\alpha)] = 1$, i.e. $\displaystyle K=F(\alpha)$.