Prove that $K=k(\alpha)$

Question

Prove:

If K\k is a Galois extension and $\alpha \in K$ with $\sigma(\alpha)\neq \alpha$ for all $\sigma \in Gal(K,k)\backslash \lbrace id_K \rbrace$, than $K=k(\alpha)$.

Answer 1

Consider $k \subseteq k(\alpha) \subseteq K$ then $K$ is Galois over $k(\alpha)$, if $\beta \in K$ is and $\sigma$ fixes $k(\alpha)$ then $\sigma =id$ so $\sigma (\beta)=\beta$, this $\beta$ is fixed by all automorphisms over $k(\alpha)$ and $\beta \in k(\alpha)$ since beta was arbitrary, $K=k(\alpha)$