Algebraic number

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In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients. Complex numbers that are not algebraic are said to be transcendental.

Contents

Examples

Properties

The field of algebraic numbers

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Numbers defined by radicals

All numbers which can be obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and taking nth 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 (which is approximately 1.167303978261418684256).

Algebraic integers

Main article: algebraic integer

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, 6i − 2 and (1 + i√3)/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 OK. These are the prototypical examples of Dedekind domains.

Special classes of algebraic number

References

  1. ^ Some of the following examples come from Hardy and Wright 1972:159-160 and pp. 178-179
  2. ^ Also Liouville's theorem can be used to "produce as many examples of transcendentals numbers as we please," cf Hardy and Wright p. 161ff
  3. ^ Hardy and Wright 1972:160
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