Definition:Algebraic Element of Field Extension

Definition
Let $E / F$ be a field extension.

Let $\alpha \in E$.

Let $f \left({x}\right)$ be a polynomial in $x$ over $F$.

Let $F \left [{X}\right]$ be the ring of polynomial forms over $F$ in the indeterminate $X$.

Then $\alpha$ is algebraic over $F$ iff:
 * $\exists f \left({x}\right) \in F \left[{X}\right] \setminus \left\{{0}\right\}: f \left({\alpha}\right) = 0$

Also see

 * An element of $E$ is said to be transcendental if it is not algebraic.


 * Algebraic number, which is the application of the definition of algebraic over $\Q$.