Definition:Field Extension

Definition
Let $F$ be a field.

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

Convention: We can write:
 * $E$ is a field extension over a field $F$

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

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

Standalone, $E/F$ means $E$ over $F$.

Degree of a Field Extension
Let $E/F$ be a field extension. Then the degree of $E/F$, denoted $[E:F]$, 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 $[E:F]< \infty$; $E/F$ is an infinite extension otherwise.