# Definition:Field Extension

## Contents

## Definition

Let $F$ be a field.

A **field extension over $F$** is a field $E$ where $F \subseteq E$.

That is, such that $F$ is a subfield of $E$.

This can be expressed:

**$E$ is a field extension over a field $F$**

or:

**$E$ over $F$ is a field extension**

as:

**$E / F$ is a field extension**.

$E / F$ can be voiced as **$E$ over $F$**.

## Degree of Field Extension

Let $E / F$ be a field extension.

The **degree of $E / F$**, denoted $\index E F$, is the dimension of $E / F$ when $E$ is viewed as a vector space over $F$.

### Finite

$E / F$ is a **finite field extension** if and only if its degree $\index E F$ is finite.

### Infinite

$E / F$ is an **infinite field extension** if and only if its degree $\index E F$ is not finite.

## Complex

Let $\left({F, +, \times}\right)$ be a subfield of $\left({\Bbb C, +, \times}\right)$, the field of complex numbers.

Let $X_1, X_2, \ldots, X_n$ be complex numbers, in or not in $F$.

Then $F \left({X_1, X_2, \ldots, X_n}\right)$ is the smallest field extension over $F$ containing $X_1, X_2, \ldots, X_n$.

## Examples

### Complex Numbers over Reals

The complex numbers $\C$ forms a **finite field extension** over the real numbers $\R$ of degree $2$.

### Numbers of Type $a + b \sqrt 2: a, b \in \Q$

Let $\Q \sqbrk {\sqrt 2}$ denote the set:

- $\Q \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Q}$

that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers.

Then $\Q \sqbrk {\sqrt 2}$ forms a **finite field extension** over the rational numbers $\Q$ of degree $2$.

## Also see

- Results about
**field extensions**can be found here.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Field Extensions: $\S 36$. The Degree of a Field Extension