Chevalley C. Introduction to the Theory of Algebraic Functions of One Variable

American Mathematical Society, 1951. - 196 pages.An algebraic function у of a complex variable χ is a function which satisfies an equation of the form F(x, y) = 0, where F is a polynomial with complex coefficients; i.e., у is a root of an algebraic equation whose coefficients are rational functions of x. This very definition exhibits a strong similarity between the notions of algebraic function and algebraic number, the rational functions of χ playing a role similar to that played by the rational numbers. On the other hand, the equation F(x, y) = 0 may be construed to represent a curve in a plane in which χ and у are the coordinates, and this establishes an intimate link between the theory of algebraic functions of one variable and algebraic geometry.