Algebraic number

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently—by clearing denominators—with integer coefficients). Numbers such as π that are not algebraic are said to be transcendental; almost all real and complex numbers are transcendental. (Here "almost all" has the sense "all but a countable set"; see Properties below.)

Examples

The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x = a/b is the root of bx-a.[1]

The quadratic surds (irrational roots of a quadratic polynomial \( ax^2 + bx + c \) with integer coefficients a, b, and c) are algebraic numbers. If the quadratic polynomial is monic (a = 1) then the roots are quadratic integers.

The constructible numbers (those that, starting with a unit length, can be constructed with straightedge and compass). These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots.

Any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number.

Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of nth roots (such as the roots of \( x^5 - x + 1 \). This happens with many, but not all, polynomials of degree 5 or higher.

Gaussian integers: those complex numbers a+bi where both a and b are integers are also quadratic integers.

Trigonometric functions of rational multiples of \pi (except when undefined). For example, each of \( cos(\pi/7), cos(3\pi/7), cos(5\pi/7) \) satisfies \( 8x^3 - 4x^2 - 4x + 1 = 0 \). This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, \( tan(3\pi/16), tan(7\pi/16), tan(11\pi/16), tan(15\pi/16) \) all satisfy the irreducible polynomial \( x^4 - 4x^3 - 6x^2 + 4x + 1 \), and so are conjugate algebraic integers.

Some irrational numbers are algebraic and some are not:
The numbers \( \sqrt{2} \) and \( \sqrt[3]{3}/2 \)are algebraic since they are roots of polynomials \( x^2 - 2 \) and \(8x^3 - 3 \), respectively.
The golden ratio \(\phi \) is algebraic since it is a root of the polynomial \( x^2 - x - 1.\)
The numbers \( \pi \) and e are not algebraic numbers (see the Lindemann–Weierstrass theorem);[2] hence they are transcendental.

Properties
Algebraic numbers coloured by degree. (red=1, green=2, blue=3, yellow=4)

The set of algebraic numbers is countable (enumerable).[3]
Hence, the set of algebraic numbers has Lebesgue measure zero (as a subset of the complex numbers), i.e. "almost all" complex numbers are not algebraic.
Given an algebraic number, there is a unique monic polynomial (with rational coefficients) 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.
All algebraic numbers are computable and therefore definable and arithmetical.
The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

The field of algebraic numbers

The sum, difference, product and quotient of two algebraic numbers is again algebraic (this fact can be demonstrated using the resultant), and the algebraic numbers therefore form a field, sometimes denoted by A (which may also denote the adele ring) or Q. 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.
Related fields
Numbers defined by radicals

All numbers which can be obtained from the integers using a finite number of integer 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 polynomial x5 − x − 1 (which is approximately 1.167304).
Closed-form number
Main article: Closed-form number

Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include algebraic numbers, but does include some simple transcendental numbers such as e or log(2).
Algebraic integers
Main article: algebraic integer
Algebraic numbers coloured by leading coefficient (red signifies 1 for an algebraic integer).

An algebraic integer is an algebraic number which is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are 5 + 13√2, 2 − 6i, and ½(1 + i√3). (Note, therefore, that the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials x − k for all k ∈ Z.)

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

Gaussian integer
Eisenstein integer
Quadratic irrational
Fundamental unit
Root of unity
Gaussian period
Pisot-Vijayaraghavan number
Salem number

Notes

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

References

Artin, Michael (1991), Algebra, Prentice Hall, ISBN 0-13-004763-5, MR1129886
Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84 (Second ed.), Berlin, New York: Springer-Verlag, ISBN 0-387-97329-X, MR1070716
G. H. Hardy and E. M. Wright 1978, 2000 (with general index) An Introduction to the Theory of Numbers: 5th Edition, Clarendon Press, Oxford UK, ISBN 0-19-853171-0
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556
Øystein Ore 1948, 1988, Number Theory and Its History, Dover Publications, Inc. New York, ISBN 0-486-65620-9 (pbk.)