TITLE: What is an algebraic function?
QUESTION [8 upvotes]: I am doing first year university calculus, and we are learning about the different kinds of functions. According to wikipedia, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients.
I understand what this means, and I get the gist of what an algebraic function typically looks like, but I'm curious, what is the formal definition of an algebraic function?

REPLY [2 votes]: Generally if $\rm\: T \supset R\:$ is ring extension then one says that $\rm\:t\in T\:$ is algebraic over $\rm\:R\:$ if $\rm\:t\:$ is the root of some polynomial $\rm\:0\ne f(x)\in R[x]\:.\:$ Some authors restrict the definition to the case of fields or domains. One calls the extension $\rm\:T\supset R\:$ algebraic if every element of $\rm\:t\:$ is algebraic over $\rm\:R\:.\:$ Elements of $\rm\:t\:$ that are not algebraic over $\rm\:R\:$ are called transcendental over $\rm\:R\:.\:$ For example, a simple extension $\rm\:R[t]\:$ is a polynomial ring  $\rm\:R[t]\:\cong R[x]\:$ iff $\rm\:t\:$ is transcendental over $\rm\:R\:.\:$ 
Algebraic "functions" are elements algebraic over a "function" ring, i.e. a polynomial ring $\rm\:R[x_1,\:\ldots,x_n]\:$ in $\rm\:n \ge 1\:$ variables. Just as one does in the case of polynomial and rational "functions", one  distinguishes between  formal algebraic "functions", and functional ones, i.e. elements of an abstract ring vs. elements of a ring whose elements are (set-theoretic) functions.