Algebraic number

Algebraic number

An algebraic number is a complex number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. All integers and rational numbers are algebraic, as are all roots of integers. The same is not true for all real and complex numbers.

Definition

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients). All integers and rational numbers are algebraic, as are all roots of integers.

Properties

The set of all algebraic numbers is countable. Therefore, almost all real and complex numbers are transcendental. The set of algebraic numbers forms a field, the algebraic numbers field, that is algebraically closed (i.e., every non-constant polynomial with algebraic coefficients has an algebraic root), and that is the smallest algebraically closed field containing the rational numbers.

Algebraic integers

An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients that is monic (i.e., the coefficient of the highest power is 1). For example, every integer is an algebraic integer.

See also

• Transcendental number
• Algebraic number field
• Algebraic integer
• Algebraically closed field