# Definition:Algebraic Number/Degree

< Definition:Algebraic Number(Redirected from Definition:Degree of Algebraic Number)

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## 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 $F$.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental