# 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 $F$** if and only if 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 $F$** if and only if 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**.