Simple extension

Definition

A field extension $L/K$ is simple if there is an element $\theta \in L$ with $L = K(\theta)$. Such an element $\theta$ is a primitive element, or a generator, of $L/K$.

Remarks

The extensions $\mathbb{C}/\mathbb{R}$, generated by $i$, and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$, generated by $\sqrt{2}$, are simple. The extension $\mathbb{Q}(\sqrt{3}, \sqrt{7})/\mathbb{Q}$ is simple, with the generator $\sqrt{3} + \sqrt{7}$. By the primitive element theorem, every finite separable extension is simple. For finite fields, the term "primitive element" usually means a generator of the multiplicative group, which is a stronger condition.

Explanations

L1 human

A simple extension is built by adding one new element to a field. Simple extensions are well understood: there is one kind for an algebraic generator and one kind for a transcendental generator.