Algebraic number

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In mathematics, an algebraic number is any number that is a solution of an algebraic equation. In other words, an algebraic number is the root of a non-zero polynomial with integer (or equivalently, rational) coefficients. Without further qualification, it is assumed that an algebraic number is a complex number, but one can also consider algebraic numbers in other fields, such as fields of p-adic numbers. All these algebraic numbers belong to some algebraic number field.

1 Examples
2 Properties
3 The field of algebraic numbers
4 Numbers defined by radicals
5 Algebraic integers
6 Special classes of algebraic number

[edit] Examples

All rationals are algebraic. An irrational number may or may not be algebraic. For example, 2^1/2 (the square root of 2) and 3^1/3/2 (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.

[edit] Properties

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 also the set of transcendental numbers, are not. Examples of transcendental numbers include π and e. Other examples are provided by the Gelfond-Schneider theorem.

All algebraic numbers are computable and therefore definable.

If an algebraic number satisfies a polynomial equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be an algebraic number 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.

[edit] 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 <math>\mathbb{A}</math> or <math>\overline{\mathbb{Q}}</math>. 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.

[edit] 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.

[edit] Algebraic integers

Main article: algebraic integer

An algebraic number which satisfies a polynomial equation of degree n with leading coefficient a_n = 1 (that is, a monic polynomial) and all other coefficients a_i belonging to the set Z of integers, is called an algebraic integer. 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.