Math Help - Field Extensions

1. Field Extensions

1. Let K be a field extension of F of prime degree p and let $\alpha$ be an element of K which is not in F. Prove that K = F( $\alpha$)

im not sure how to solve this

2. We have $p=[K:F]=[K:F(\alpha)]\times[F(\alpha):F]$. Thus $[F(\alpha):F] = p$ or 1. Since $F(\alpha)\supsetneq F$, we can’t have $[F(\alpha):F] = 1$; therefore $[F(\alpha):F] = p$ $\Rightarrow$ $[K:F(\alpha)] = 1$, i.e. $K=F(\alpha)$.