In [[mathematics]], an '''algebraic number''' is a [[complex number]] that is a [[root (mathematics)|root]] of a non-zero [[polynomial]] with [[rational number|rational]] (or equivalently, [[integer]]) coefficients. Complex numbers that are not algebraic are said to be [[transcendental number|transcendental]].
==Examples==
*The [[rational number]]s, those expressed as the ratio of two whole numbers ''b'' and ''a'', ''a'' not equal to zero, satisfy the above definition because x = -b/a is derived from (and satisfies) a*x + b = 0. (In general, ''a'' or ''b'' can be negative, as can x).<ref> Some of the following examples come from Hardy and Wright 1972:159-160 and pp. 178-179</ref>
* When the lead coefficent e.g. a<sub>0</sub> is 1, the satisfactory x is/are said to be (an) [[algebraic integer]]/s. Note that an "algebraic integer" need not be a counting number such as 1, 2, 3, ... or a negative counterpart.
:* This definition comes from the notion that x = -b/a satisfies ''a''*x + ''b'' = 0, and when ''a'' = 1 then x = -''b'' (i.e. ''b'' here being a positive or negative counting number or 0). But observe that from 1*x<sup>2</sup> + 4 = 0, x = 2i and -2i. So these two x are "algebraic integers" as well. This applies for any value of lead-exponent n. (See more below).
*The [[quadratic surd]]s (roots of a quadratic equation ax<sup>2</sup> + bx + c = 0 with integral coefficents a, b, and c) are algebraic numbers. Thus those complex numbers derived from ax<sup>2</sup> + bx + c=0 -- those corresponding to the case when the exponent n = 2 -- are called [[quadratic number]]s, or [[quadratic integer]]s as the case may be.
:* [[Gaussian integer]]s -- those complex numbers a + bi with "rational-integer" a and b -- are also quadratic integers. Here, a "rational integer" is what we commonly think of as just "an integer" (aka whole number), e.g. a/c with c = 1 and b/d with d = 1.
:* The [[golden ratio]] φ is algebraic since it is a root of the polynomial ''x''<sup>2</sup> − ''x'' − 1 = 0.
:* The [[irrational number]]s <math>\sqrt{2}</math> and <math>\sqrt[3]{3}/2</math> are algebraic since they are the roots of ''x''<sup>2</sup> − 2 = 0 and 8''x''<sup>3</sup> − 3 = 0, respectively.
* The [[Euclidean number]]s (those that, starting with a unit, can be constructed with ruler and compass, e.g. the square root of 2) are algebraic. A less-simple example involves the real square roots of the sum of an integer and a square root.
* The real numbers [[Pi|<math>\pi</math>]] and [[e (mathematical constant)|<math>e</math>]] are '''not''' algebraic numbers (see the [[Lindemann–Weierstrass theorem]])<ref>Also [[Liouville's theorem]] can be used to "produce as many examples of transcendentals numbers as we please," cf Hardy and Wright p. 161ff</ref> hence they are "transcendental".
:* The fact that most numbers, indeed almost all, are "transcendental" is proven by use of the Cantor [[diagonal method]].<ref>Hardy and Wright 1972:160</ref>
==Properties==
* The set of algebraic numbers is [[countable set|countable]] (enumerable).<ref>Hardy and Wright 1972:160</ref>
* Given an algebraic number, there is a unique [[monic polynomial]] (with rational coefficients) of least [[polynomial|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 number|computable]] and therefore [[definable number|definable]].
==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 (mathematics)|field]], sometimes denoted by <math>\mathbb{A}</math> (which may also denote the [[adele ring]]) 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 field|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 set|finite]] number of [[addition]]s, [[subtraction]]s, [[multiplication]]s, [[division (mathematics)|division]]s, and taking ''n''<sup>th</sup> 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 equation]]s and the [[Abel–Ruffini theorem]]). An example of such a number is the unique real root of ''x''<sup>5</sup> − x − 1 = 0.
==Algebraic integers== {{main|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√{{overline|2}} + 5, 6''i'' - 2 and (1+''i''√{{overline|3}})/2.
The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a [[ring (algebra)|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 [[algebraic number field|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''<sub>K</sub>.
These are the prototypical examples of [[Dedekind domain]]s.
==Special classes of algebraic number==
*[[Gaussian integer]]
*[[Eisenstein integer]]
*[[Quadratic irrational]]
*[[Fundamental unit (number theory)|Fundamental unit]]
*[[Root of unity]]
*[[Gaussian period]]
*[[Pisot-Vijayaraghavan number]]
*[[Salem number]]
== Footnotes ==
{{reference}}
== References ==
* {{Citation | last=Artin | first=Michael | author-link=Michael Artin | title=Algebra | publisher=[[Prentice Hall]] | isbn=0-13-004763-5 | id={{MathSciNet | id = 1129886}} | year=1991}}
* {{Citation | last1=Ireland | first1=Kenneth | last2=Rosen | first2=Michael | title=A Classical Introduction to Modern Number Theory | edition=Second | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=0-387-97329-X | id={{MathSciNet | id = 1070716}} | year=1990 | volume=84}}
* [[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
* [[Orestein Ore]] 1948, 1988, ''Number Theory and Its History'', Dover Publications, Inc. New York, ISBN 0-486-65620-9 (pbk.)
[[Category:Abstract algebra]]
[[Category:Algebra]]
[[Category:Algebraic numbers|*]]
[[Category:Number theory]]
[[Category:Algebraic number theory]]
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