Primitive element theorem

Statement

Suppose $L/K$ is a finite extension of fields, i.e., the degree of the extension is finite. The primitive element theorem states that there exists an element $\alpha \in L$ such that $L=K(\alpha )$. In other words, the extension can be generated by a single element over $K$. Such an element is termed a primitive element.

Related facts

• Normal basis theorem states that any Galois extension has a basis (as a vector space over the base field) that forms a single orbit under the action of the Galois group.