Algebraic number

An algebraic number is any complex number that is a root of a polynomial with rational coefficients. Any polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators, and every complex root of a polynomial with integer coefficients is an algebraic number. If an algebraic number x can be written as the root of a monic polynomial, that is, one whose leading coefficient is 1, then x is called an algebraic integer.

The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are countable. The algebraic numbers form a field; in fact, they are the smallest algebraically closed field with characteristic 0.

Real or complex numbers that are not algebraic are called transcendental numbers.

Examples

${\sqrt {2}}$ is an algebraic number, and, in fact, an algebraic integer, as it is a root of the polynomial $x^{2}-2$. Similarly, the imaginary unit $i$ is an algebraic integer, being a root of the polynomial $x^{2}+1$.

Algebraic numbers via subalgebras and subfields

The field of complex numbers $\ \mathbb {C}$  is a linear space over the field of rational numbers $\ \mathbb {Q} .$  In this section, by a linear space we will mean a linear subspace of $\ \mathbb {C}$  over $\ \mathbb {Q} ;$  and by algebra we mean a linear space which is closed under the multiplication, and which has $\ 1$  as its element. The following properties of a complex number $\ z\in \mathbb {C}$  are equivalent:

• $\ z$  is an algebraic number of degree $\ \leq n;$
• $\ z$  belongs to an algebra of linear dimension $\ \leq n.$

Indeed, when the first condition holds, then the powers $\ 1,z,\dots ,z^{n-1}$  linearly generate the algebra required by the second condition. And if the second condition holds then the $\ (n+1)$  elements 1,z,\dots,z^{n}</math>  are linearly dependent (over rationals).

Actually, every finite dimensional algebra $\ A\subseteq \mathbb {C}$  is a field—indeed, divide an equality

$a_{0}\cdot z^{n}+\dots +a_{n-1}\cdot z+a_{n}\ =\ 0$

where $\ a_{0}\neq 0\neq a_{n},$  by $\ a_{n}\cdot z,$  and you quickly get an equality of the form:

$z^{-1}\ =\ b_{0}\cdot z^{n-1}+\cdots +b_{n-1}$