Algebraic number

In mathematics, an algebraic number is a complex number that is an algebraic element over the rational numbers. In other words, an algebraic number is a root of a non-zero polynomial with rational (or equivalently, integer) coefficients.

Examples

All rational numbers are algebraic. An irrational number may or may not be algebraic. For example, $\textstyle {\sqrt {2}}$ (the square root of 2) and $\textstyle {\frac {1}{2}}{\sqrt[{3}]{3}}$ (half the cube root of 3) are algebraic because they are the solutions of x² − 2 = 0 and 8x³ − 3 = 0, respectively. The imaginary unit i is algebraic, since it satisfies x² + 1 = 0. The most common examples of numbers which are not algebraic are π and e (see the Lindemann–Weierstrass theorem).

Properties

Complex numbers that are not algebraic are called transcendental numbers. Most complex numbers are transcendental, because the set of algebraic numbers is countable while the set of complex numbers, and therefore the set of transcendental numbers, is not (since the union of two countable sets is countable). Other examples of transcendental numbers are provided by the Gelfond-Schneider theorem.

All algebraic numbers are computable and therefore definable.

Given an algebraic number, there is a unique monic polynomial of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number.

The concept of algebraic numbers can be generalized to arbitrary field extensions; elements in such extensions that satisfy polynomial equations are called algebraic elements.

The field of algebraic numbers

The sum, difference, product and quotient of two algebraic numbers is again algebraic, and the algebraic numbers therefore form a field, sometimes denoted by $\mathbb {A}$ (which may also denote the adele ring) or ${\overline {\mathbb {Q} }}$ . It can be shown that every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. This can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

All the above statements are most easily proved in the general context of algebraic elements of a field extension.

Numbers defined by radicals

All numbers which can be obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and taking n^th roots (where n is a positive integer) are algebraic. The converse, however, is not true: there are algebraic numbers which cannot be obtained in this manner. All of these numbers are solutions to polynomials of degree ≥ 5. This is a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). An example of such a number is the unique real root of x⁵ − x − 1 = 0.

==Algebraic integers==

An algebraic integer is a number which is a root of a polynomial with integer coefficients (that is, an algebraic number) with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are 3√2 + 5 and 6i - 2.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring. The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O_K. These are the prototypical examples of Dedekind domains.

Special classes of algebraic number