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.

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. ^[1]

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

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 $x^{2}+1$ .

Notes

↑ If 1 + 1 = 0 the characteistic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no $n$ such that adding 1 $n$ times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite.

[1] If 1 + 1 = 0 the characteistic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no $n$ such that adding 1 $n$ times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite.

[1]

Algebraic number

Examples

Notes

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