# Definition:Algebraic Number over Field

## Definition

Let $F$ be a field.

Let $z$ be a complex number.

$z$ is algebraic over $F$ if and only if $z$ is a root of a polynomial with coefficients in $F$.

### Degree

Let $F$ be a field.

Let $z \in \C$ be algebraic over $F$.

The degree of $\alpha$ is the degree of the minimal polynomial $\map m x$ whose coefficients are all in $F$.