# Algebraic Element of Field Extension/Definition 1

## Definition

Let $E / F$ be a field extension.

Let $\alpha \in E$.

$\alpha$ is algebraic over $F$ if and only if 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 $X$.