Simple extension

Let $E/F$ be a field extension and let $\alpha\in E$. The simple extension generated by $\alpha$ is the smallest subfield of $E$ containing both $F$ and $\alpha$; it is denoted

Equivalently,

If $E=F(\alpha)$, then $E/F$ is called a simple extension.

A useful description is: $F(\alpha)$ consists of all rational expressions in $\alpha$ with coefficients in $F$, i.e.

If $\alpha$ is an algebraic element over $F$, then $F(\alpha)/F$ is an algebraic extension and has finite degree . If $\alpha$ is transcendental over $F$, then $F(\alpha)/F$ is a transcendental extension .

Simple extensions are the building blocks of finitely generated field extensions : by definition, $F(\alpha)$ is the finitely generated case with one generator.

Examples

1. $\mathbb{Q}(\sqrt2)/\mathbb{Q}$ is simple: $\mathbb{Q}(\sqrt2)$ is the smallest field containing $\mathbb{Q}$ and $\sqrt2$.
2. $\mathbb{C}/\mathbb{R}$ is simple: $\mathbb{C}=\mathbb{R}(i)$.
3. If $t$ is an indeterminate, then $\mathbb{Q}(t)/\mathbb{Q}$ is simple: it is generated (as a field) by the transcendental element $t$.