Algebraic number

An algebraic number is a complex number that is a possible root of a non-zero polynomial function with rational coefficients and natural exponents.

Definition: The set of all algebraic numbers
$\mathbb {A} =\left\{\alpha \in \mathbb {C} \mid \exists P\in \mathbb {Q} \left[x\right]:P\left(\alpha \right)=0,P\left(x\right)\neq 0{\text{ for some }}x\right\}$ where $\mathbb {Q} \left[x\right]$ is the set of all polynomial functions with rational coefficients.

For example, if we take $\alpha ={\sqrt {3}}$, $\beta =2i$, and $\gamma ={\sqrt {2}}+{\sqrt[{3}]{3}}$, we will find these numbers to be algebraic since:

$\alpha ^{2}-3=0$;

$\beta ^{2}+4=0$;

$\gamma ^{6}-6\gamma ^{4}-6\gamma ^{3}+12\gamma ^{2}-36\gamma +1=0$.

Also, any rational number $q$ is also algebraic, since it is a root of $x-q$.

Any number that is not algebraic, that is, not a root of any non-trivial polynomial function with rational roots, is defined to be transcendental.

In this way, constructing such a polynomial with a specific root proves that the given root is an algebraic number.

Algebraic numbers are either rational or irrational numbers, purely real or imaginary, or a complex combination. Essentially, most all complex numbers conceivable through simple algebraic relationships are algebraic, while the transcendentals exist between them and algebraically unrelated to them.

See also

• Annihilating an algebraic number with a polynomial with rational coefficients