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 roots 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 smallest $n$ such that $n$ is the degree of such a polynomial $P_n$.


Examples

The algebraic numbers include:

The elements of the set of rational numbers $\Q$, since these are roots of linear polynomials -- see Rational Number is Algebraic
$\sqrt 2$, since it is a root of $x^2 - 2$
$\sqrt [3] {2 + \sqrt 2}$, since it is a root of $x^6 - 4 x^3 + 2$
$\sqrt 3 + \sqrt 2$, since it is a root of $x^4 - 10 x^2 + 1$
$\sqrt [3] 4 - 2 i$, since it is a root of $x^6 + 12 x^4 - 8 x^3 + 48 x^2 + 96 x + 80$
The golden section $\varphi$, since it is a root of $x^2 - x - 1$
The imaginary unit $i$, since it is a root of $x^2 + 1$


Also see


Examples


Generalizations


Sources