Editing Algebraic function (section)

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{{Short description|Mathematical function}}
In [[mathematics]], an '''algebraic function''' is a [[Function (mathematics)|function]] that can be defined 
as the [[Zero of a function|root]] of an [[Irreducible polynomial|irreducible]] [[polynomial equation]]. Algebraic functions are often [[algebraic expression]]s using a finite number of terms, involving only the [[algebraic operations]] addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:
* <math>f(x) = 1/x</math>
* <math>f(x) = \sqrt{x}</math>
* <math>f(x) = \frac{\sqrt{1 + x^3}}{x^{3/7} - \sqrt{7} x^{1/3}}</math>

Some algebraic functions, however, cannot be expressed by such finite expressions (this is the [[Abel–Ruffini theorem]]). This is the case, for example, for the [[Bring radical]], which is the function [[implicit function|implicitly]] defined by

: <math>f(x)^5+f(x)+x = 0</math>.

In more precise terms, an algebraic function of degree {{math|''n''}} in one variable {{math|''x''}} is a function <math>y = f(x),</math> that is [[Continuous function|continuous]] in its [[domain of a function|domain]] and satisfies a [[polynomial equation]] of positive [[degree of a polynomial |degree]]

: <math>a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0</math>

where the coefficients {{math|''a''<sub>''i''</sub>(''x'')}} are [[polynomial function]]s of {{math|''x''}}, with integer coefficients. It can be shown that the same class of functions is obtained if [[algebraic numbers]] are accepted for the coefficients of the {{math|''a''<sub>''i''</sub>(''x'')}}'s. If [[transcendental number]]s occur in the coefficients the function is, in general, not algebraic, but it is ''algebraic over the [[Field (mathematics)|field]]'' generated by these coefficients.

The value of an algebraic function at a [[rational number]], and more generally, at an [[algebraic number]] is always an algebraic number.
Sometimes, coefficients <math>a_i(x)</math> that are polynomial over a [[Ring (mathematics)|ring]] {{mvar|R}} are considered, and one then talks about "functions algebraic over {{mvar|R}}".

A function which is not algebraic is called a [[transcendental function]], as it is for example the case of <math>\exp x, \tan x, \ln x, \Gamma(x)</math>. A composition of transcendental functions can give an algebraic function:  <math>f(x)=\cos \arcsin x = \sqrt{1-x^2}</math>.

As a polynomial equation of [[Degree of a polynomial|degree]] ''n'' has up to ''n'' roots (and exactly ''n'' roots over an [[algebraically closed field]], such as the [[complex numbers]]), a polynomial equation does not implicitly define a single function, but up to ''n''
functions, sometimes also called [[branch cut|branches]]. Consider for example the equation of the [[unit circle]]:
<math>y^2+x^2=1.\,</math>
This determines ''y'', except only [[up to]] an overall sign; accordingly, it has two branches:
<math>y=\pm \sqrt{1-x^2}.\,</math>

An '''algebraic function in ''m'' variables''' is similarly defined as a function <math>y=f(x_1,\dots ,x_m)</math> which solves a polynomial equation in ''m''&thinsp;+&thinsp;1 variables:

:<math>p(y,x_1,x_2,\dots,x_m) = 0.</math>

It is normally assumed that ''p'' should be an [[irreducible polynomial]].  The existence of an algebraic function is then guaranteed by the [[implicit function theorem]].

Formally, an algebraic function in ''m'' variables over the field ''K'' is an element of the [[algebraic closure]] of the field of [[rational function]]s ''K''(''x''<sub>1</sub>,&thinsp;...,&thinsp;''x''<sub>''m''</sub>).