Primitive Element Theorem

Theorem (Primitive Element).
If $L/K$ is a finite separable extension , then $L$ is a simple extension of $K$: there exists $\alpha\in L$ such that

Equivalently, every finite separable extension is generated by a single element.

Examples.

1. $\mathbb{Q}(\sqrt2,\sqrt3) = \mathbb{Q}(\sqrt2+\sqrt3)$.
   (One checks $\sqrt2,\sqrt3 \in \mathbb{Q}(\sqrt2+\sqrt3)$.)
2. For any prime power $q=p^n$, the extension $\mathbb{F}_{q}/\mathbb{F}_p$ is simple: $\mathbb{F}_{q}=\mathbb{F}_p(\alpha)$ for some $\alpha$.
3. The splitting field $L$ of $x^3-2$ over $\mathbb{Q}$ is finite separable (characteristic $0$), hence $L=\mathbb{Q}(\alpha)$ for some $\alpha\in L$.

Related. field extensions , perfect ⇒ separable .