Normal basis theorem

Statement

Suppose $L/K$ is a finite Galois extension of fields with Galois group $G$. Then, there exists an element $\alpha \in L$ such that the set:

$\!\{\sigma (\alpha )\mid \sigma \in G\}$

forms a basis for $L$ as a vector space over $K$. In other words, we can always find a basis that is a single orbit under the action of the Galois group. Such a basis is termed a normal basis.