Definition:Algebraic Number

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Definition

An algebraic number is an algebraic element of the field extension $\C / \Q$.

That is, it is a complex number that is a root of a polynomial with rational coefficients.


The set of algebraic number can often be seen denoted as $\mathbb A$.


Degree

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

Some sources define an algebraic number over a more general field:


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 $F$.


Examples

The algebraic numbers include the following:


Rational Number is Algebraic

Let $r \in \Q$ be a rational number.

Then $r$ is also an algebraic number.


$\sqrt 2$ is Algebraic

$\sqrt 2$ is an algebraic number.


$\sqrt {2 + \sqrt 3}$ is Algebraic

$\sqrt {2 + \sqrt 3}$ is an algebraic number.


$\sqrt [3] {2 + \sqrt 2}$ is Algebraic

$\sqrt [3] {2 + \sqrt 2}$ is an algebraic number.


$\sqrt [3] 2 + \sqrt 3$ is Algebraic

$\sqrt [3] 2 + \sqrt 3$ is an algebraic number.


$\sqrt 3 + \sqrt 2$ is Algebraic

$\sqrt 3 + \sqrt 2$ is an algebraic number.


$2 - \sqrt 2 i$ is Algebraic

$2 - \sqrt 2 i$ is an algebraic number.


$\sqrt [3] 4 - 2 i$ is Algebraic

$\sqrt [3] 4 - 2 i$ is an algebraic number.


Golden Mean is Algebraic

The golden mean $\phi = \dfrac {1 + \sqrt 5} 2$ is an algebraic number.


Imaginary Unit is Algebraic

The imaginary unit $i$ is an algebraic number.


Also see

  • Results about algebraic numbers can be found here.


Sources

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