---
tag: 0P6SBR7YM2
kind: definition
name: Simple extension
msc: [12F05]
defines: 0N681SSTHC
uses: [0MA5B744K3]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_extension
    locator: Definition
---

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ 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|0SVX98HYV1]], every finite separable extension is simple.
For finite fields, the term "primitive element" usually means a [[generator of the multiplicative group|0YAA9CJQXF]], 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.