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.