Algebraic element

Let $𝕂$ be a field and $𝐴$ be a $𝕂$ -monoid (or extension field, see extension field as a unital associative algebra).

An element $𝑎 \in 𝐴$ is called algebraic over $𝕂$ iff there exists a nonzero polynomial $𝑝 (𝑥) \in 𝕂 [𝑥]$ such that $𝑝 (𝑎) = 0$ . #m/def/ralg An element which is not algebraic is called transcendental over $𝕂$ . If $𝑎$ is algebraic, the solving monic polynomial of smallest degree $𝑚 𝑎 (𝑥) \in 𝕂 [𝑥]$ is called the minimal polynomial of $𝑎$ . This is a special case of Integral element, and thus the set is denoted $O 𝐴 : 𝕂$

$𝐴$ is called algebraic over $𝕂$ iff every $𝑎 \in 𝐴$ is algebraic, and if $𝐴$ is a field the field extension $𝐴 : 𝕂$ is called algebraic.

An algebraic element over $ℚ$ is called an algebraic number.

Examples

Number field
All elements of a finite-dimensional unital associative algebra are algebraic

Properties

Subalgebra generated by an algebraic element
An algebraic element is invertible iff its minimal polynomial has a nonzero constant term
Roots of a minimal polynomial
Spectrum of an algebraic element
Embedding an algebraic extension into an algebraically closed field

Constructions

Algebraic interior of a field extension

#state/tidy | #lang/en | #SemBr