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 $S'$ such that:

$F \subseteq S'$

$\theta_1, \theta_2, \ldots, \theta_n \in S'$

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

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 36$. The Degree of a Field Extension