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, so an algebraic number is also a root of a polynomial with integer coefficients.

The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are countable. The algebraic numbers for 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 numberss.

Examples

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