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Algebraic number

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From Wikipedia, the free encyclopedia

Type of complex number

Not to be confused withAlgebraic solution.

The square root of 2 is an algebraic number equal to the length of thehypotenuse of aright triangle with legs of length 1.

Inmathematics, analgebraic number is a number that is aroot of a non-zeropolynomial in one variable withinteger (or, equivalently,rational) coefficients. For example, thegolden ratio $(1+{\sqrt {5}})/2$ is an algebraic number, because it is a root of the polynomial $x^{2}-x-1$ , i.e., a solution to the equation $x^{2}-x-1=0$ , and thecomplex number $1+i$ is algebraic because it is a root of the polynomial $x^{4}+4$ . Algebraic numbers include allintegers,rational numbers, andn-th roots of integers.

Algebraiccomplex numbers are closed under addition, subtraction, multiplication and division, and hence form afield, denoted ${\overline {\mathbb {Q} }}$ . The set of algebraicreal numbers ${\overline {\mathbb {Q} }}\cap \mathbb {R}$ is also a field.

Numbers which are not algebraic are calledtranscendental and includeπ ande. There arecountably infinite algebraic numbers, hencealmost all real (or complex) numbers (in the sense ofLebesgue measure) are transcendental.

Examples

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Allrational numbers are algebraic. Any rational number, expressed as the quotient of anintegera and a (non-zero)natural numberb, satisfies the above definition, becausex =⁠a/b⁠ is the root of a non-zero polynomial, namelybx −a.^[1]
Quadratic irrational numbers, irrational solutions of a quadratic polynomialax² +bx +c with integer coefficientsa,b, andc, are algebraic numbers. If the quadratic polynomial is monic (a = 1), the roots are further qualified asquadratic integers.
- Gaussian integers, complex numbersa +bi for which botha andb are integers, are also quadratic integers. This is becausea +bi anda −bi are the two roots of the quadraticx² − 2ax +a² +b².
Aconstructible 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 thebasic arithmetic operations and the extraction of square roots. (By designating cardinal directions for +1, −1, +i, and −i, complex numbers such as $3+i{\sqrt {2}}$ are considered constructible.)
Any expression formed from algebraic numbers using any finite combination of the basic arithmetic operations and extraction ofnth roots gives another algebraic number.
Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction ofnth roots (such as the roots ofx⁵ −x + 1).That happens with many but not all polynomials of degree 5 or higher.
Values oftrigonometric functions of rational multiples ofπ (except when undefined): for example,cos⁠π/7⁠,cos⁠3π/7⁠, andcos⁠5π/7⁠ satisfy8x³ − 4x² − 4x + 1 = 0. This polynomial isirreducible over the rationals and so the three cosines areconjugate algebraic numbers. Likewise,tan⁠3π/16⁠,tan⁠7π/16⁠,tan⁠11π/16⁠, andtan⁠15π/16⁠ satisfy the irreducible polynomialx⁴ − 4x³ − 6x² + 4x + 1 = 0, and so are conjugatealgebraic integers. This is the equivalent of angles which, when measured in degrees, have rational numbers.^[2]
Some but not all irrational numbers are algebraic:
- The numbers ${\sqrt {2}}$ and ${\frac {\sqrt[{3}]{3}}{2}}$ are algebraic since they are roots of polynomialsx² − 2 and8x³ − 3, respectively.
- Thegolden ratioφ is algebraic since it is a root of the polynomialx² −x − 1.
- The numbersπ ande are not algebraic numbers (see theLindemann–Weierstrass theorem).^[3]

Properties

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Algebraic numbers on thecomplex 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 theleast 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 uniquemonic polynomial with rational coefficients of leastdegree that has the number as a root. This polynomial is called itsminimal polynomial. If its minimal polynomial has degreen, then the algebraic number is said to be ofdegreen. For example, allrational numbers have degree 1, and an algebraic number of degree 2 is aquadratic irrational.
The algebraic numbers aredense 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,^[4]^[5] and therefore itsLebesgue 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 all" real and complex numbers are transcendental.
All algebraic numbers arecomputable and thereforedefinable andarithmetical.
For real numbersa andb, the complex numbera +bi is algebraic if and only if botha andb are algebraic.^[6]

Degree of simple extensions of the rationals as a criterion to algebraicity

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For any⁠ $\alpha$ ⁠, thesimple extension of the rationals by⁠ $\alpha$ ⁠, denoted by $\mathbb {Q} (\alpha )$ (whose elements are the $f(\alpha )$ for $f {\displaystyle f}$ arational function with rational coefficients which is defined at $\alpha$ ), is of finitedegree if and only if⁠ $\alpha$ ⁠ is an algebraic number.

The condition of finite degree means that there is a fixed set of numbers $\{a_{i}\}$ of finitecardinality⁠ $k {\displaystyle k}$ ⁠ with elements in $\mathbb {Q} (\alpha )$ such that $\textstyle \mathbb {Q} (\alpha )=\sum _{i=1}^{k}a_{i}\mathbb {Q}$ ; that is, each element of $\mathbb {Q} (\alpha )$ can be written as a sum $\textstyle \sum _{i=1}^{k}a_{i}q_{i}$ for some rational coefficients $\{q_{i}\}$ .

Since the $a_{i}$ are themselves members of $\mathbb {Q} (\alpha )$ , each can be expressed as sums of products of rational numbers and powers of⁠ $\alpha$ ⁠, and therefore this condition is equivalent to the requirement that for some finite $n {\displaystyle n}$ , $\mathbb {Q} (\alpha )={\biggl \lbrace }\sum _{i=-n}^{n}\alpha ^{i}q_{i}\mathbin {\bigg |} q_{i}\in \mathbb {Q} {\biggr \rbrace }.$

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

It can similarly be proven that for any finite set of algebraic numbers $\alpha _{1}$ , $\alpha _{2}$ ... $\alpha _{n}$ , the field extension $\mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})$ has a finite degree.

Field

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Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black.^{[further explanation needed]}

The sum, difference, product, and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic:

For any two algebraic numbers⁠ $\alpha$ ⁠,⁠ $\beta$ ⁠, this follows directly from the fact that thesimple extension $\mathbb {Q} (\gamma )$ , for $\gamma$ being either $\alpha +\beta$ , $\alpha -\beta$ , $\alpha \beta$ or (for $\beta \neq 0$ ) $\alpha /\beta$ , is alinear subspace of the finite-degree field extension $\mathbb {Q} (\alpha ,\beta )$ , and therefore has a finite degree itself, from which it follows (as shownabove) that $\gamma$ is algebraic.

An alternative way of showing this is constructively, by using theresultant.

Algebraic numbers thus form afield^[7] ${\overline {\mathbb {Q} }}$ (sometimes denoted by $\mathbb {A}$ , but that usually denotes theadele ring).

Algebraic closure

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Every root of a polynomial equation whose coefficients arealgebraic numbers is again algebraic. That can be rephrased by saying that the field of algebraic numbers isalgebraically closed. In fact, it is the smallest algebraically closed field containing the rationals and so it is called thealgebraic closure of the rationals.

That the field of algebraic numbers is algebraically closed can be proven as follows: Let⁠ $\beta$ ⁠ be a root of a polynomial $\alpha _{0}+\alpha _{1}x+\alpha _{2}x^{2}...+\alpha _{n}x^{n}$ with coefficients that are algebraic numbers $\alpha _{0}$ , $\alpha _{1}$ , $\alpha _{2}$ ... $\alpha _{n}$ . The field extension $\mathbb {Q} ^{\prime }\equiv \mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})$ then has a finite degree with respect to $\mathbb {Q}$ . The simple extension $\mathbb {Q} ^{\prime }(\beta )$ then has a finite degree with respect to $\mathbb {Q} ^{\prime }$ (since all powers of⁠ $\beta$ ⁠ can be expressed by powers of up to $\beta ^{n-1}$ ). Therefore, $\mathbb {Q} ^{\prime }(\beta )=\mathbb {Q} (\beta ,\alpha _{1},\alpha _{2},...\alpha _{n})$ also has a finite degree with respect to $\mathbb {Q}$ . Since $\mathbb {Q} (\beta )$ is a linear subspace of $\mathbb {Q} ^{\prime }(\beta )$ , it must also have a finite degree with respect to $\mathbb {Q}$ , so⁠ $\beta$ ⁠ must be an algebraic number.

Related fields

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Numbers defined by radicals

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Any number that can be obtained from the integers using afinite number ofadditions,subtractions,multiplications,divisions, and taking (possibly complex)nth roots wheren 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 5 or higher, a result ofGalois theory (seeQuintic equations and theAbel–Ruffini theorem). For example, the equation:

x^{5}-x-1=0

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

Closed-form number

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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 numbersexplicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such ase orln 2.

Algebraic integers

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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 article:Algebraic integer

Analgebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (amonic polynomial). Examples of algebraic integers are $5+13{\sqrt {2}},$ $2-6i,$ and ${\textstyle {\frac {1}{2}}(1+i{\sqrt {3}}).}$ Therefore, the algebraic integers constitute a propersuperset of theintegers, as the latter are the roots of monic polynomialsx −k for all $k\in \mathbb {Z}$ . In this sense, algebraic integers are to algebraic numbers whatintegers are torational numbers.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form aring. The namealgebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in anynumber field are in many ways analogous to the integers. IfK is a number field, itsring of integers is the subring of algebraic integers inK, and is frequently denoted asO_K. These are the prototypical examples ofDedekind domains.

Special classes

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Notes

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^Some of the following examples come fromHardy & Wright (1972, pp. 159–160, 178–179)
^Garibaldi 2008.
^Also,Liouville's theorem can be used to "produce as many examples of transcendental numbers as we please," cf.Hardy & Wright (1972, p. 161ff)
^Hardy & Wright 1972, p. 160, 2008:205.
^Niven 1956, Theorem 7.5..
^Niven 1956, Corollary 7.3..
^Niven 1956, p. 92.

References

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Artin, Michael (1991),Algebra, Prentice Hall,ISBN 0-13-004763-5,MR 1129886
Garibaldi, Skip (June 2008), "Somewhat more than governors need to know about trigonometry",Mathematics Magazine,81 (3):191–200,doi:10.1080/0025570x.2008.11953548,JSTOR 27643106
Hardy, Godfrey Harold;Wright, Edward M. (1972),An introduction to the theory of numbers (5th ed.), Oxford: Clarendon,ISBN 0-19-853171-0
Ireland, Kenneth; Rosen, Michael (1990) [1st ed. 1982],A Classical Introduction to Modern Number Theory (2nd ed.), Berlin: Springer,doi:10.1007/978-1-4757-2103-4,ISBN 0-387-97329-X,MR 1070716
Lang, Serge (2002) [1st ed. 1965],Algebra (3rd ed.), New York: Springer,ISBN 978-0-387-95385-4,MR 1878556
Niven, Ivan M. (1956),Irrational Numbers,Mathematical Association of America
Ore, Øystein (1948),Number Theory and Its History, New York: McGraw-Hill

v t e Algebraic numbers
Algebraic integer Chebyshev nodes Constructible number Conway's constant Cyclotomic field Doubling the cube Eisenstein integer Gaussian integer Golden ratio (φ) Perron number Pisot–Vijayaraghavan number Plastic ratio (ρ) Quadratic irrational number Rational number Root of unity Salem number Silver ratio (σ) Square root of 2 Square root of 3 Square root of 5 Square root of 6 Square root of 7 Square root of 10 Supergolden ratio (ψ) Supersilver ratio (ς) Twelfth root of 2
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v t e Number systems
Sets ofdefinable numbers	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers ( $\mathbb {A}$ ) Closed-form numbers Periods ( ${\mathcal {P}}$ ) Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers Gaussian rationals Eisenstein integers
Composition algebras	Division algebras:Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ ) Quaternions ( $\mathbb {H}$ ) Octonions ( $\mathbb {O}$ )
Split types	Over $\mathbb {R}$ : Split-complex numbers Split-quaternions Split-octonions Over $\mathbb {C}$ : Bicomplex numbers Biquaternions Bioctonions
Otherhypercomplex	Dual numbers Dual quaternions Dual-complex numbers Hyperbolic quaternions Sedenions ( $\mathbb {S}$ ) Trigintaduonions ( $\mathbb {T}$ ) Split-biquaternions Multicomplex numbers Geometric algebra/Clifford algebra Algebra of physical space Spacetime algebra Plane-based geometric algebra
Infinities andinfinitesimals	Cardinal numbers Extended natural numbers Extended real numbers Projective Extended complex numbers Hyperreal numbers Levi-Civita field Ordinal numbers Supernatural numbers Surreal numbers Superreal numbers
Other types	Irrational numbers Fuzzy numbers Transcendental numbers p-adic numbers (p-adic solenoids) Profinite integers Normal numbers
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