# Definition:Algebraic Number/Degree

## Definition

Let $\alpha$ be an algebraic number.

By definition, $\alpha$ is the root of at least one polynomial $P_n$ with rational coefficients.

The degree of $\alpha$ is the degree of the minimal polynomial $P_n$ whose coefficients are all in $\Q$.

### Algebraic Number over Field

Sources which define an algebraic number over a more general field define degree in the following terms:

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 $F$.