Algebraic number

Not to be confused withAlgebraic solution.

The square root of 2 is an algebraic number equal to the length of thehypotenuseof aright trianglewith legs of length 1.

Analgebraic numberis a number that is arootof a non-zeropolynomial(of finite degree) 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 polynomialx²−x− 1.That is, it is a value for x for which the polynomial evaluates to zero. As another example, thecomplex number$1+i$is algebraic because it is a root ofx⁴+ 4.

All integers and rational numbers are algebraic, as are allroots of integers.Real and complex numbers that are not algebraic, such asπande,are calledtranscendental numbers.

Thesetof algebraic numbers iscountably infiniteand hasmeasure zeroin theLebesgue measureas asubsetof theuncountablecomplex numbers. In that sense,almost allcomplex numbers aretranscendental.

Examples

• Allrational numbersare algebraic. Any rational number, expressed as the quotient of anintegeraand a (non-zero)natural numberb,satisfies the above definition, becausex=a/bis the root of a non-zero polynomial, namelybx−a.
• Quadratic irrational numbers,irrational solutions of a quadratic polynomialax²+bx+cwith 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+bifor which bothaandbare integers, are also quadratic integers. This is becausea+bianda−biare the two roots of the quadraticx²− 2ax+a²+b².
• Aconstructible numbercan 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 operationsand 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 combination of the basic arithmetic operations and extraction ofnth rootsgives 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 manybut not all polynomials of degree 5 or higher.
• Values oftrigonometric functionsof rational multiples ofπ(except when undefined): for example,cosπ/7,cos3π/7,andcos5π/7satisfy8x³− 4x²− 4x+ 1 = 0.This polynomial isirreducibleover the rationals and so the three cosines areconjugatealgebraic numbers. Likewise,tan3π/16,tan7π/16,tan11π/16,andtan15π/16satisfy 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.
• 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− 2and8x³− 3,respectively.
  • Thegolden ratioφis algebraic since it is a root of the polynomialx²−x− 1.
  • The numbersπandeare not algebraic numbers (see theLindemann–Weierstrass theorem).

Properties

Algebraic numbers on thecomplex planecolored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4)

• 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 polynomialwith rational coefficients of leastdegreethat 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 numbershave degree 1, and an algebraic number of degree 2 is aquadratic irrational.
• The algebraic numbers aredensein 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 (enumerable),and therefore itsLebesgue measureas 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 arecomputableand thereforedefinableandarithmetical.
• For real numbersaandb,the complex numbera+biis algebraic if and only if bothaandbare algebraic.

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

For anyα,thesimple extensionof the rationals byα,denoted by$\mathbb {Q} (\alpha )\equiv \{\sum _{i=-{n_{1}}}^{n_{2}}\alpha ^{i}q_{i}|q_{i}\in \mathbb {Q},n_{1},n_{2}\in \mathbb {N} \}$,is of finitedegreeif and only ifαis an algebraic number.

The condition of finite degree means that there is a finite set$\{a_{i}|1\leq i\leq k\}$in$\mathbb {Q} (\alpha )$such that$\mathbb {Q} (\alpha )=\sum _{i=1}^{k}a_{i}\mathbb {Q}$;that is, every member in$\mathbb {Q} (\alpha )$can be written as$\sum _{i=1}^{k}a_{i}q_{i}$for some rational numbers$\{q_{i}|1\leq i\leq k\}$(note that the set$\{a_{i}\}$is fixed).

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

The latter condition is equivalent to$\alpha ^{n+1}$,itself a member of$\mathbb {Q} (\alpha )$,being expressible as$\sum _{i=-n}^{n}\alpha ^{i}q_{i}$for some rationals$\{q_{i}\}$,so$\alpha ^{2n+1}=\sum _{i=0}^{2n}\alpha ^{i}q_{i-n}$or, equivalently,αis a root of$x^{2n+1}-\sum _{i=0}^{2n}x^{i}q_{i-n}$;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

Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black.

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

For any two algebraic numbersα,β,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 subspaceof the finite-degreefield 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${\overline {\mathbb {Q} }}$(sometimes denoted by$\mathbb {A}$,but that usually denotes theadele ring).

Algebraic closure

Every root of a polynomial equation whose coefficients arealgebraic numbersis 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 closureof the rationals.

That the field of algebraic numbers is algebraically closed can be proven as follows: Letβbe 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 ofβcan 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$\mathbb {Q}$,soβmust be an algebraic number.

Related fields

Numbers defined by radicals

Any number that can be obtained from the integers using afinitenumber ofadditions,subtractions,multiplications,divisions,and taking (possibly complex)nth roots wherenis 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 equationsand theAbel–Ruffini theorem). For example, the equation:

$x^{5}-x-1=0$

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

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 numbersexplicitlydefined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such aseorln 2.

Algebraic integers

Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer)

Main article:Algebraic integer

Analgebraic integeris 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}}).}$Therefore, the algebraic integers constitute a propersupersetof theintegers,as the latter are the roots of monic polynomialsx−kfor all$k\in \mathbb {Z}$.In this sense, algebraic integers are to algebraic numbers whatintegersare torational numbers.

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

Special classes

• Algebraic solution
• Gaussian integer
• Eisenstein integer
• Quadratic irrational number
• Fundamental unit
• Root of unity
• Gaussian period
• Pisot–Vijayaraghavan number
• Salem number

References

• Artin, Michael(1991),Algebra,Prentice Hall,ISBN0-13-004763-5,MR1129886
• 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,JSTOR27643106
• Hardy, Godfrey Harold;Wright, Edward M.(1972),An introduction to the theory of numbers(5th ed.), Oxford: Clarendon,ISBN0-19-853171-0
• 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,ISBN0-387-97329-X,MR1070716
• Lang, Serge (2002) [1st ed. 1965],Algebra(3rd ed.), New York: Springer,ISBN978-0-387-95385-4,MR1878556
• Niven, Ivan M.(1956),Irrational Numbers,Mathematical Association of America
• Ore, Øystein(1948),Number Theory and Its History,New York: McGraw-Hill