Definition:Field Extension

Definition

Let $F$ be a field.

Then 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 a Field Extension

Let $E/F$ be a field extension.

Then the degree of $E/F$, denoted $\left[{E:F}\right]$, is the dimension of $E/F$ when $E$ is viewed as a vector space over $F$.

We say $E/F$ is a finite extension if $\left[{E:F}\right] < \infty$; $E/F$ is an infinite extension otherwise.

Also see

• Results about field extensions can be found here.

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous): $\S 8.36$