Algebraic function

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.

Common examples of such functions are:

Some algebraic functions, however, cannot be expressed by such finite expressions (this is Abel–Ruffini theorem). This is the case, for example, of 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 ${\displaystyle y=f(x)}$ that satisfies a polynomial equation

where the coefficients a_i(x) are polynomial functions of x, with coefficients belonging to a set S. Quite often, ${\displaystyle S=\mathbb {Q} }$, and one then talks about "function algebraic over ${\displaystyle \mathbb {Q} }$", and the evaluation at a given rational value of such an algebraic function gives an algebraic number.

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