Definition:Finitely Generated Field Extension
(Redirected from Definition:Finitely Generated Extension)
Jump to navigation
Jump to search
Definition
Let $E / F$ be a field extension.
Then $E$ is said to be finitely generated over $F$ if and only if, for some $\alpha_1, \ldots, \alpha_n \in E$:
- $E = F \left({\alpha_1, \ldots, \alpha_n}\right)$
where $F \left({\alpha_1, \ldots, \alpha_n}\right)$ is the field in $E$ generated by $F \cup \left\{{\alpha_1, \ldots, \alpha_n}\right\}$.