Some basic definitions and fundamental properties of algebraic function fields are introduced in this chapter. In particular, we focus on those concepts and results that are needed in the subsequent chapters, such as the Riemann-Roch theorem, divisor class groups, Galois extensions of algebraic function fields, ramification theory, constant field extensions, zeta functions, and the Hasse-Weil bound.

In later chapters, we will be interested only in algebraic function fields over finite fields, but a lot of the background material can be developed for arbitrary constant fields. In this chapter, we will not always state results in their most general form, but only in the form in which we need them. Most results will be presented here without proof since they are standard results from textbooks. The reader will find the books of Stichtenoth [152] and Weiss [173] particularly useful. We refer also to the books of Cassels and Fröhlich [13], Deuring [18], Koch [60], Moreno [82], Neukirch [85], and Stepanov [150], [151] for further background.