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 $F$.

Also see

• Definition:Transcendental Number over Field

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): algebraic: 3 b.
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 38$. Simple Algebraic Extensions