# Definition:Smallest Field containing Subfield and Complex Number/General Definition

## Definition

Let $F$ be a field.

Let $\theta_1, \theta_2, \ldots, \theta_n \in \C$ be complex numbers.

Let $S$ be the intersection of all fields such that:

$S \subseteq F$
$\theta_1, \theta_2, \ldots, \theta_n \in F$

Then $S$ is denoted $\map F {\theta_1, \theta_2, \ldots, \theta_n}$ and referred to as the smallest field containing $F$ and $\theta_1, \theta_2, \ldots, \theta_n$.