\begin{definition}[Definition:Algebraic Element of Field Extension/Definition 1]
Let $E / F$ be a field extension.
Let $\alpha \in E$.
$\alpha$ is '''algebraic over $F$''' {{iff}} it is a root of some nonzero polynomial over $F$:
:$\exists f \in F \sqbrk X \setminus \set 0: \map f \alpha = 0$
where $F \sqbrk X$ denotes the ring of polynomial forms in $X$.
\end{definition}