Algebraic function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an algebraic function is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that can be defined 
as the <a href="/facts/Zero_of_a_function/mlK3dhoT">root</a> of an <a href="/facts/Irreducible_polynomial/PTeuKm4W">irreducible</a> <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equation</a>. Algebraic functions are often <a href="/facts/Algebraic_expression/CK7D57Dm">algebraic expressions</a> using a finite number of terms, involving only the <a href="/facts/Algebraic_operations/wl69LmeN">algebraic operations</a> addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

<ul><li>
 
 
 
 f
 (
 x
 )
 =
 1
 
 /
 
 x
 
 
 {\displaystyle f(x)=1/x}
 
</li>
<li>
 
 
 
 f
 (
 x
 )
 =
 
 
 x
 
 
 
 
 {\displaystyle f(x)={\sqrt {x}}}
 
</li>
<li>
 
 
 
 f
 (
 x
 )
 =
 
 
 
 1
 +
 
 x
 
 3
 
 
 
 
 
 x
 
 3
 
 /
 
 7
 
 
 −
 
 
 7
 
 
 
 x
 
 1
 
 /
 
 3
 
 
 
 
 
 
 
 {\displaystyle f(x)={\frac {\sqrt {1+x^{3}}}{x^{3/7}-{\sqrt {7}}x^{1/3}}}}
 
</li></ul>
Some algebraic functions, however, cannot be expressed by such finite expressions (this is the <a href="/facts/Abel%E2%80%93Ruffini_theorem/tEFaRGvW">Abel–Ruffini theorem</a>). This is the case, for example, for the <a href="/facts/Bring_radical/uAgB4rHe">Bring radical</a>, which is the function <a href="/facts/Implicit_function/UJmQ2Cyk">implicitly</a> defined by

f
 (
 x
 
 )
 
 5
 
 
 +
 f
 (
 x
 )
 +
 x
 =
 0
 
 
 {\displaystyle f(x)^{5}+f(x)+x=0}
 
.
In more precise terms, an algebraic function of degree n in one variable x is a function 
 
 
 
 y
 =
 f
 (
 x
 )
 ,
 
 
 {\displaystyle y=f(x),}
 
 that is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> in its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> and satisfies a <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equation</a> of positive <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a>

a
          
            n
          
        
        (
        x
        )
        
          y
          
            n
          
        
        +
        
          a
          
            n
            −
            1
          
        
        (
        x
        )
        
          y
          
            n
            −
            1
          
        
        +
        ⋯
        +
        
          a
          
            0
          
        
        (
        x
        )
        =
        0
      
    
    {\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0}

where the coefficients ai(x) are <a href="/facts/Polynomial_function/Lzak8VVx">polynomial functions</a> of x, with integer coefficients. It can be shown that the same class of functions is obtained if <a href="/facts/Algebraic_numbers/MkH1PzTK">algebraic numbers</a> are accepted for the coefficients of the ai(x)'s. If <a href="/facts/Transcendental_number/IrB7JppM">transcendental numbers</a> occur in the coefficients the function is, in general, not algebraic, but it is algebraic over the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> generated by these coefficients.
The value of an algebraic function at a <a href="/facts/Rational_number/VJQmyY05">rational number</a>, and more generally, at an <a href="/facts/Algebraic_number/MkH1PzTK">algebraic number</a> is always an algebraic number.
Sometimes, coefficients 
 
 
 
 
 a
 
 i
 
 
 (
 x
 )
 
 
 {\displaystyle a_{i}(x)}
 
 that are polynomial over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R are considered, and one then talks about "functions algebraic over R".
A function which is not algebraic is called a <a href="/facts/Transcendental_function/yMkeq4Eo">transcendental function</a>, as it is for example the case of 
 
 
 
 exp
 ⁡
 x
 ,
 tan
 ⁡
 x
 ,
 ln
 ⁡
 x
 ,
 Γ
 (
 x
 )
 
 
 {\displaystyle \exp x,\tan x,\ln x,\Gamma (x)}
 
. A composition of transcendental functions can give an algebraic function: 
 
 
 
 f
 (
 x
 )
 =
 cos
 ⁡
 arcsin
 ⁡
 x
 =
 
 
 1
 −
 
 x
 
 2
 
 
 
 
 
 
 {\displaystyle f(x)=\cos \arcsin x={\sqrt {1-x^{2}}}}
 
.
As a polynomial equation of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> n has up to n roots (and exactly n roots over an <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a>, such as the <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>), a polynomial equation does not implicitly define a single function, but up to n
functions, sometimes also called <a href="/facts/Branch_cut/lJ7Y6Iht">branches</a>. Consider for example the equation of the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a>:

y
          
            2
          
        
        +
        
          x
          
            2
          
        
        =
        1.
        
      
    
    {\displaystyle y^{2}+x^{2}=1.\,}

This determines y, except only <a href="/facts/Up_to/ydz4ztzX">up to</a> an overall sign; accordingly, it has two branches:

y
        =
        ±
        
          
            1
            −
            
              x
              
                2
              
            
          
        
        .
        
      
    
    {\displaystyle y=\pm {\sqrt {1-x^{2}}}.\,}

An algebraic function in m variables is similarly defined as a function 
 
 
 
 y
 =
 f
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 m
 
 
 )
 
 
 {\displaystyle y=f(x_{1},\dots ,x_{m})}
 
 which solves a polynomial equation in m + 1 variables:

p
        (
        y
        ,
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        ,
        
          x
          
            m
          
        
        )
        =
        0.
      
    
    {\displaystyle p(y,x_{1},x_{2},\dots ,x_{m})=0.}

It is normally assumed that p should be an <a href="/facts/Irreducible_polynomial/PTeuKm4W">irreducible polynomial</a>. The existence of an algebraic function is then guaranteed by the <a href="/facts/Implicit_function_theorem/r4h0RpWw">implicit function theorem</a>.
Formally, an algebraic function in m variables over the field K is an element of the <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of the field of <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a> K(x1, ..., xm).

Algebraic function open-in-new

Algebraic function