Definition:Algebraic Element of Field Extension

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

Let $\alpha \in E$.

Definition 1
The element $\alpha$ is algebraic over $F$ it is a root of some nonzero polynomial over $F$:
 * $\exists f \in F \left[{X}\right] \setminus \left\{{0}\right\}: f \left({\alpha}\right) = 0$

Definition 2
The element $\alpha$ is algebraic over $F$ the evaluation homomorphism $F[x] \to K$ at $\alpha$ is not injective.

Also see

 * Definition:Algebraic Field Extension
 * Definition:Algebraic Independence
 * Definition:Integral Element of Algebra
 * An element of $E$ is said to be transcendental if it is not algebraic.

Special cases

 * Definition:Algebraic Number