INTRODUCTION An algebraic function y of a complex variable x is a function which satisfies an equation of the form F(x, y) = 0, where F is a polynomial with complex coefficients i.e., y 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 x 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 x and y are the coordinates, and this establishes an intimate link between the theory of algebraic functions of one variable and algebraic geometry. Whoever wants to give an exposition of the theory of algebraic functions of one variable is more or less bound to lay more emphasis either on the algebraico- arithmetic aspect of this branch of mathematics or on its geometric aspect. Both points of view are acceptable and have been in fact held by various mathe- maticians. The algebraic attitude was first distinctly asserted in the paper Theorie der algebraischen Funktionen einer Verdnderlichen, by R. Dedekind and H. Weber (Journ. fur Math., 92, 1882, pp. 181-290), and inspires the book Theorie der algebraischen Funktionen einer Variabeln, by Hensel and Landsberg (Leipzig, 1902). The geometric approach was followed by Max Noether, Clebsch, Gordan, and, after them, by the geometers of the Italian school (cf. in particular the book Lezione di Geometria algebrica, by F. Severi, Padova, 1908). Whichever method is adopted, the main results to be established are of course essentially the same but this common material is made to reflect a different light when treated by differently minded mathematicians. Familiar as we are with the idea that the pair "observed fact—observer" is probably a more real being than the inert fact or theorem by itself, we shall not neglect the diversity of these various angles under which a theory may be photographed. Such a neglect should be particularly avoided in the case of the theory of algebraic functions, as either mode of approach seems liable to provoke strong emotional reactions in mathe- matical minds, ranging from devout enthusiasm to unconditional rejection. However, this does not mean that the ideal should consist in a mixture or synthe- sis of the two attitudes in the writing of any one book: the only result of trying to obtain two interesting photographs of the same object on the same plate is a blurred and dull image. Thus, without attacking in any way the validity per se of the geometric approach, we have not tried to hide our partiality to the algebraic attitude, which has been ours in writing this book. The main difference between the present treatment of the theory and the one to be found in Dedekind-Weber or in Hensel-Landsberg lies in the fact that the constants of the fields of algebraic functions to be considered are not necessarily the complex numbers, but the elements of a completely arbitrary field. There ix
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