[[Ring theory MOC]]
# Algebraic element

Let $\mathbb{K}$ be a [[field]] and $A$ be a [[K-monoid]] (or [[Field extension|extension field]], see [[extension field as a unital associative algebra]]).

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

$A$ is called algebraic over $\mathbb{K}$ iff every $a \in A$ is algebraic,
and if $A$ is a field the [[field extension]] $A : \mathbb{K}$ is called **algebraic**.

An algebraic element over $\mathbb{Q}$ is called an [[Number field|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]]

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