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 $F$.

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