A field $E$ is a field extension of the field $F$ if $F$ is a subfield of $E$.
The field extension may be considered as a vector space over $F$. The dimension of $E$ as an $F$-vector space is denoted $[E:F]$, $\deg(E/F)$, or $\deg_F(E)$.
The most basic field extension comes from adjoining an element $\Theta$ to $F$. It is denoted $F(\Theta)$, and it is the smallest field containing both $F$ and $\Theta$.