\begin{definition}[Definition:Field Adjoined Element]
{{delete|think it fits more naturally here}}
Let $E/F$ be a field extension, $\alpha \in E$.
Then:
:$F[\alpha] $ denotes the smallest subring of $E$ containing $F \cup \alpha$.
:$F(\alpha) $ denotes the smallest subfield of $E$ containing $F \cup \alpha$.  We say this as '''$F$ adjoined with $\alpha$'''.
Category:Definitions/Field Theory
\end{definition}