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Concept
Algebraic function
Summary
In mathematics, an algebraic function is a function that can be defined
as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions 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, which is the function implicitly defined by
In more precise terms, an algebraic function of degree n in one variable x is a function that is continuous in its domain and satisfies a polynomial equation
where the coefficients ai(x) are polynomial functions of 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 ai(x)'s. If transcendental numbers 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, and more generally, at an algebraic number is always an algebraic number.
Sometimes, coefficients that are polynomial over a ring R are considered, and one then talks about "functions algebraic over R".
A function which is not algebraic is called a transcendental function, 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, such as the complex numbers), a polynomial equation does not implicitly define a single function, but up to n
functions, sometimes also called branches. Consider for example the equation of the unit circle:
This determines y, except only up to 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.
Official source
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