Algebraic function - All About All

Algebraic function

In mathematics, an algebraic function of indeterminates

X₁, X₂, ..., X_n,

is a function F that satisfies some non-trivial equation

P(F, X₁, X₂, ..., X_n) = 0,

with P a polynomial in n + 1 variables over a given field K. That is, F is an implicit function that solves an algebraic equation. A simple example would be

F(X) = √(X² + 1).

The class of algebraic functions contains all rational functions, but is larger. In fact in terms of abstract algebra it is the algebraic closure of the field of rational functions, for any fixed set of indeterminates. (NB: when K is a finite field it is inaccurate to equate polynomials with functions, rather than symbolic expressions; however the term algebraic function is used nonetheless).