View source for Algebraic number

{{Short description|Type of complex number}}
{{Distinguish|Algebraic solution}}
{{Use shortened footnotes|date=September 2024}}
[[File:Isosceles right triangle with legs length 1.svg|thumb|200px|The square root of 2 is an algebraic number equal to the length of the [[hypotenuse]] of a [[right triangle]] with legs of length 1.]]

In [[mathematics]], an '''algebraic number''' is a number that is a [[root of a function|root]] of a non-zero [[polynomial]] in one variable with [[integer]] (or, equivalently, [[Rational number|rational]]) coefficients.  For example, the [[golden ratio]] <math>(1 + \sqrt{5})/2</math> is an algebraic number, because it is a root of the polynomial <math>x^2 - x - 1</math>, i.e., a solution to the equation <math>x^2 - x - 1 = 0</math>, and the [[complex number]] <math>1 + i</math> is algebraic because it is a root of the polynomial <math>x^4 + 4</math>. Algebraic numbers include all [[integer]]s, [[rational number]]s, and [[nth root|''n''-th roots of integers]].

Algebraic [[complex number]]s are closed under addition, subtraction, multiplication and division, and hence form a [[field (mathematics)|field]], denoted <math>\overline{\Q}</math>. The set of algebraic [[real number]]s <math>\overline{\Q} \cap \R</math> is also a field.

Numbers which are not algebraic are called [[transcendental number|transcendental]] and include [[pi|{{pi}}]] and {{mvar|[[e (mathematical constant)|e]]}}. There are [[countable set|countably infinite]] algebraic numbers, hence [[almost all]] real (or complex) numbers (in the sense of [[Lebesgue measure]]) are transcendental.

==Examples==
* All [[rational number]]s are algebraic. Any rational number, expressed as the quotient of an [[integer]] {{mvar|a}} and a (non-zero) [[natural number]] {{mvar|b}}, satisfies the above definition, because {{math|''x'' {{=}} {{sfrac|''a''|''b''}}}} is the root of a non-zero polynomial, namely {{math|''bx'' − ''a''}}.<ref>Some of the following examples come from {{harvtxt|Hardy|Wright|1972|pp=159–160, 178–179}}</ref>
* [[Quadratic irrational number]]s, irrational solutions of a quadratic polynomial {{math|''ax''{{sup|2}} + ''bx'' + ''c''}} with integer coefficients {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}, are algebraic numbers. If the quadratic polynomial is monic ({{math|''a'' {{=}} 1}}), the roots are further qualified as [[quadratic integer]]s.
** [[Gaussian integer]]s, complex numbers {{math|''a'' + ''bi''}} for which both {{mvar|a}} and {{mvar|b}} are integers, are also quadratic integers. This is because {{math|''a'' + ''bi''}} and {{math|''a'' − ''bi''}} are the two roots of the quadratic {{math|''x''{{sup|2}} − 2''ax'' + ''a''{{sup|2}} + ''b''{{sup|2}}}}.
* A [[constructible number]] can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the [[Arithmetic operations|basic arithmetic operations]] and the extraction of square roots. (By designating cardinal directions for +1, −1, +{{mvar|i}}, and −{{mvar|i}}, complex numbers such as <math>3+i \sqrt{2}</math> are considered constructible.)
* Any expression formed from algebraic numbers using any finite combination of the basic arithmetic operations and extraction of [[nth root|{{mvar|n}}th roots]] gives another algebraic number.
* Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of {{mvar|n}}th roots (such as the roots of {{math|''x''<sup>5</sup> − ''x'' + 1}}). [[Abel–Ruffini theorem|That happens with many]] but not all polynomials of degree 5 or higher.
* Values of [[trigonometric functions]] of rational multiples of {{pi}} (except when undefined): for example, {{math|cos {{sfrac|{{math|π}}|7}}}}, {{math|cos {{sfrac|3{{math|π}}|7}}}}, and {{math|cos {{sfrac|5{{math|π}}|7}}}} satisfy {{math|8''x''<sup>3</sup> − 4''x''<sup>2</sup> − 4''x'' + 1 {{=}} 0}}. This polynomial is [[irreducible polynomial|irreducible]] over the rationals and so the three cosines are ''conjugate'' algebraic numbers. Likewise, {{math|tan {{sfrac|3{{math|π}}|16}}}}, {{math|tan {{sfrac|7{{math|π}}|16}}}}, {{math|tan {{sfrac|11{{math|π}}|16}}}}, and {{math|tan {{sfrac|15{{math|π}}|16}}}} satisfy the irreducible polynomial {{math|''x''<sup>4</sup> − 4''x''<sup>3</sup> − 6''x''<sup>2</sup> + 4''x'' + 1 {{=}} 0}}, and so are conjugate [[algebraic integer]]s. This is the equivalent of angles which, when measured in degrees, have rational numbers.{{sfn|Garibaldi|2008}}
* Some but not all irrational numbers are algebraic:
** The numbers <math>\sqrt{2}</math> and <math>\frac{ \sqrt[3]{3} }{ 2 }</math> are algebraic since they are roots of polynomials {{math|''x''<sup>2</sup> − 2}} and {{math|8''x''<sup>3</sup> − 3}}, respectively.
** The [[golden ratio]] {{mvar|φ}} is algebraic since it is a root of the polynomial {{math|''x''<sup>2</sup> − ''x'' − 1}}.
** The numbers [[pi|{{pi}}]] and [[e (mathematical constant)|e]] are not algebraic numbers (see the [[Lindemann–Weierstrass theorem]]).<ref>Also, [[Liouville number|Liouville's theorem]] can be used to "produce as many examples of transcendental numbers as we please," cf. {{harvtxt|Hardy|Wright|1972|p=161ff}}</ref>

==<span class="anchor" id="Degree of an algebraic number"></span> Properties==
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[[File:Algebraicszoom.png|thumb|Algebraic numbers on the [[complex plane]] colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4). The larger points come from polynomials with smaller integer coefficients.]]
*If a polynomial with rational coefficients is multiplied through by the [[least common denominator]], the resulting polynomial with integer coefficients has the same roots.  This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients.
*Given an algebraic number, there is a unique [[monic polynomial]] with rational coefficients of least [[degree of a polynomial|degree]] that has the number as a root. This polynomial is called its [[minimal polynomial (field theory)|minimal polynomial]]. If its minimal polynomial has degree {{mvar|n}}, then the algebraic number is said to be of '''degree {{mvar|n}}'''. For example, all [[rational number]]s have degree 1, and an algebraic number of degree 2 is a [[quadratic irrational]].
*The algebraic numbers are [[dense set|dense]] [[densely ordered|in the reals]].  This follows from the fact they contain the rational numbers, which are dense in the reals themselves.
*The set of algebraic numbers is countable,{{sfn|Hardy|Wright|1972|p=160|loc=2008:205}}{{sfn|Niven|1956|loc=Theorem 7.5.}} and therefore its [[Lebesgue measure]] as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, [[Almost everywhere|"almost all"]] real and complex numbers are transcendental.
*All algebraic numbers are [[computable number|computable]] and therefore [[definable number|definable]] and [[arithmetical numbers|arithmetical]].
*For real numbers {{math|''a''}} and {{math|''b''}}, the complex number {{math|''a'' + ''bi''}} is algebraic if and only if both {{math|''a''}} and {{math|''b''}} are algebraic.{{sfn|Niven|1956|loc=Corollary 7.3.}}

===Degree of simple extensions of the rationals as a criterion to algebraicity===
For any {{tmath|\alpha}}, the [[simple extension]] of the rationals by {{tmath|\alpha}}, denoted by <math>\Q(\alpha)</math> (whose elements are the <math>f(\alpha)</math> for <math>f</math> a [[rational function]] with rational coefficients which is defined at <math>\alpha</math>), is of finite [[Degree of a field extension|degree]] if and only if {{tmath|\alpha}} is an algebraic number.

The condition of finite degree means that there is a fixed set of numbers <math>\{a_i\}</math> of finite [[cardinality]] {{tmath|k}} with elements in <math>\Q(\alpha)</math> such that <math>\textstyle \Q(\alpha) = \sum_{i=1}^k a_i \Q</math>; that is, each element of <math>\Q(\alpha)</math> can be written as a sum <math>\textstyle \sum_{i=1}^k a_i q_i</math> for some rational coefficients <math>\{q_i \}</math>.

Since the <math>a_i</math> are themselves members of <math>\Q(\alpha)</math>, each can be expressed as sums of products of rational numbers and powers of {{tmath|\alpha}}, and therefore this condition is equivalent to the requirement that for some finite <math>n</math>, <math display=block>\Q(\alpha) = \biggl\lbrace \sum_{i=-n}^n \alpha^{i} q_i \mathbin{\bigg|} q_i\in \Q\biggr\rbrace.</math>

The latter condition is equivalent to <math>\alpha^{n+1}</math>, itself a member of <math>\Q(\alpha)</math>, being expressible as <math>\textstyle \sum_{i=-n}^n \alpha^i q_i</math> for some rationals <math>\{q_i\}</math>, so <math>\textstyle \alpha^{2n+1} = \sum_{i=0}^{2n} \alpha^i q_{i-n}</math> or, equivalently, {{tmath|\alpha}} is a root of <math>\textstyle x^{2n+1}-\sum_{i=0}^{2n} x^i q_{i-n}</math>; that is, an algebraic number with a minimal polynomial of degree not larger than <math>2n+1</math>.

It can similarly be proven that for any finite set of algebraic numbers <math>\alpha_1</math>, <math>\alpha_2</math>... <math>\alpha_n</math>, the field extension <math>\Q(\alpha_1, \alpha_2, ... \alpha_n)</math> has a finite degree.

==Field==
[[File:Algebraic number in the complex plane.png|thumb|Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black.{{explain|reason=What does this figure tell us about algebraic numbers? Can we get some insight out of it, or it this just mathematical art?|date=July 2024}}]]
The sum, difference, product, and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic:

For any two algebraic numbers {{tmath|\alpha}}, {{tmath|\beta}}, this follows directly from the fact that the [[simple extension]] <math>\Q(\gamma)</math>, for <math>\gamma</math> being either <math>\alpha+\beta</math>, <math>\alpha-\beta</math>, <math>\alpha\beta</math> or (for <math>\beta\ne 0</math>) <math>\alpha/\beta</math>, is a [[linear subspace]] of the finite-[[Degree of a field extension|degree]] field extension <math>\Q(\alpha,\beta)</math>, and therefore has a finite degree itself, from which it follows (as shown [[#Degree of simple extensions of the rationals as a criterion to algebraicity|above]]) that <math>\gamma</math> is algebraic.

An alternative way of showing this is constructively, by using the [[resultant]].

Algebraic numbers thus form a [[field (mathematics)|field]]{{sfn|Niven|1956|p=92}} <math>\overline{\mathbb{Q}}</math> (sometimes denoted by <math>\mathbb A</math>, but that usually denotes the [[adele ring]]).

===Algebraic closure===
Every root of a polynomial equation whose coefficients are ''algebraic numbers'' is again algebraic. That 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 so it is called the [[algebraic closure]] of the rationals.

That the field of algebraic numbers is algebraically closed can be proven as follows: Let {{tmath|\beta}} be a root of a polynomial <math> \alpha_0 + \alpha_1 x + \alpha_2 x^2 ... +\alpha_n x^n</math> with coefficients that are algebraic numbers <math>\alpha_0</math>, <math>\alpha_1</math>, <math>\alpha_2</math>... <math>\alpha_n</math>. The field extension <math>\Q^\prime \equiv \Q(\alpha_1, \alpha_2, ... \alpha_n)</math> then has a finite degree with respect to <math>\Q</math>. The simple extension <math>\Q^\prime(\beta)</math> then has a finite degree with respect to <math>\Q^\prime</math> (since all powers of {{tmath|\beta}} can be expressed by powers of up to <math>\beta^{n-1}</math>). Therefore, <math>\Q^\prime(\beta) = \Q(\beta, \alpha_1, \alpha_2, ... \alpha_n)</math> also has a finite degree with respect to <math>\Q</math>. Since <math>\Q(\beta)</math> is a linear subspace of <math>\Q^\prime(\beta)</math>, it must also have a finite degree with respect to <math>\Q</math>, so {{tmath|\beta}} must be an algebraic number.

==Related fields==

===Numbers defined by radicals===
Any number that 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 (possibly complex) {{mvar|n}}th roots where {{mvar|n}} is a positive integer are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree&nbsp;5 or higher, a result of [[Galois theory]] (see [[Quintic equation]]s and the [[Abel–Ruffini theorem]]). For example, the equation:

:<math>x^5-x-1=0</math>

has a unique real root, ≈ 1.1673, that cannot be expressed in terms of only radicals and arithmetic operations.

===Closed-form number===
{{Main|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 number]]s", 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 number]]s", 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 all algebraic numbers, but does include some simple transcendental numbers such as {{mvar|e}} or [[Natural logarithm of 2|ln&nbsp;2]].

==Algebraic integers==
[[Image:Leadingcoeff.png|thumb|Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top.]]
{{Main|Algebraic integer}}
An ''algebraic integer'' is an algebraic number that is a root of  a polynomial with integer coefficients with leading coefficient 1 (a [[monic polynomial]]). Examples of algebraic integers are <math>5 + 13 \sqrt{2},</math> <math>2 - 6i,</math> and <math display=inline>\frac{1}{2}(1+i\sqrt{3}).</math> Therefore, the algebraic integers constitute a proper [[superset]] of the [[integer]]s, as the latter are the roots of monic polynomials {{math|''x'' − ''k''}} for all <math>k \in \mathbb{Z}</math>. In this sense, algebraic integers are to algebraic numbers what [[integer]]s are to [[rational number]]s.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a [[Ring (mathematics)|ring]]. The name ''algebraic integer'' comes from the fact that the only rational numbers that 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 {{math|''K''}} is a number field, its [[ring of integers]] is the subring of algebraic integers in {{math|''K''}}, and is frequently denoted as {{math|''O<sub>K</sub>''}}. These are the prototypical examples of [[Dedekind domain]]s.

==Special classes==
*[[Algebraic solution]]
*[[Gaussian integer]]
*[[Eisenstein integer]]
*[[Quadratic irrational number]]
*[[Fundamental unit (number theory)|Fundamental unit]]
*[[Root of unity]]
*[[Gaussian period]]
*[[Pisot–Vijayaraghavan number]]
*[[Salem number]]

==Notes==
{{Reflist}}

==References==
*{{citation |last=Artin |first=Michael |author-link=Michael Artin |year=1991 |title=Algebra |publisher=Prentice Hall |isbn=0-13-004763-5 |mr=1129886 |url=https://archive.org/details/algebra0000arti_x4a1/ |url-access=limited }}
*{{citation
 | last = Garibaldi | first = Skip
 | date = June 2008
 | doi = 10.1080/0025570x.2008.11953548
 | issue = 3
 | journal = Mathematics Magazine
 | jstor = 27643106
 | pages = 191–200
 | title = Somewhat more than governors need to know about trigonometry
 | volume = 81}}
*{{citation |last1=Hardy |first1=Godfrey Harold |author-link1=G. H. Hardy |last2=Wright |first2=Edward M. |author-link2=E. M. Wright |date=1972 |title=An introduction to the theory of numbers |edition=5th  |location=Oxford |publisher=Clarendon|isbn=0-19-853171-0}}
*{{citation |last1=Ireland |first1=Kenneth |last2=Rosen |first2=Michael |year=1990 |orig-year=1st ed. 1982 |title=A Classical Introduction to Modern Number Theory |edition=2nd |place=Berlin |publisher=Springer |isbn=0-387-97329-X |mr=1070716 |doi=10.1007/978-1-4757-2103-4}}
*{{citation |last=Lang |first=Serge |year=2002 |orig-year=1st ed. 1965 |title=Algebra |edition=3rd |place=New York |publisher=Springer |isbn=978-0-387-95385-4 |mr=1878556 |url=https://archive.org/details/algebra-serge-lang/ }}
*{{citation |last=Niven |first=Ivan M. |author-link=Ivan M. Niven |year=1956 |title=Irrational Numbers |publisher=[[Mathematical Association of America]] |url=https://archive.org/details/irrationalnumber00nive/ |url-access=limited }}
*{{citation|last=Ore |first=Øystein |author-link=Øystein Ore |year=1948 |title=Number Theory and Its History |publisher=McGraw-Hill |location=New York |url=https://archive.org/details/numbertheoryitsh00ore/ |url-access=limited }}

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