1 Answers
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 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 y = f , {\displaystyle y=f,} that is continuous in its domain and satisfies a polynomial equation
where the coefficients ai 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'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.