In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an algebraic function is a
function that can be defined
as the
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of a
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
. Quite often algebraic functions are
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
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:
*
*
*
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
In algebra, the Bring radical or ultraradical of a real number ''a'' is the unique real root of the polynomial
: x^5 + x + a.
The Bring radical of a complex number ''a'' is either any of the five roots of the above polynomial (it is thu ...
, which is the function
implicitly defined by
:
.
In more precise terms, an algebraic function of degree in one variable is a function
that is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
in its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
and satisfies a
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
:
where the coefficients are
polynomial function
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s of , 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 's. If
transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
s occur in the coefficients the function is, in general, not algebraic, but it is ''algebraic over the
field'' generated by these coefficients.
The value of an algebraic function at a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
, and more generally, at an
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
is always an algebraic number.
Sometimes, coefficients
that are polynomial over a
ring are considered, and one then talks about "functions algebraic over ".
A function which is not algebraic is called a
transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed ...
, as it is for example the case of
. A composition of transcendental functions can give an algebraic function:
.
As a polynomial equation of
degree ''n'' has up to ''n'' roots (and exactly ''n'' roots over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
, such as the
complex numbers), a polynomial equation does not implicitly define a single function, but up to ''n''
functions, sometimes also called
branches
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term ''twig'' usually r ...
. Consider for example the equation of the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
:
This determines ''y'', except only
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
an overall sign; accordingly, it has two branches:
An algebraic function in ''m'' variables is similarly defined as a function
which solves a polynomial equation in ''m'' + 1 variables:
:
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
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s ''K''(''x''
1, ..., ''x''
''m'').
Algebraic functions in one variable
Introduction and overview
The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual
algebraic operations:
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
,
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
,
division, and taking an
''n''th root. This is something of an oversimplification; because of the
fundamental theorem of Galois theory
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In its most basi ...
, algebraic functions need not be expressible by radicals.
First, note that any
polynomial function
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
is an algebraic function, since it is simply the solution ''y'' to the equation
:
More generally, any
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
is algebraic, being the solution to
:
Moreover, the ''n''th root of any polynomial