Definition:Algebraic Number/Degree
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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