ALGEBRAIC FUNCTION FIELDS OVER FINITE FIELDS

Algebraic function fields are important in several areas of information theory such as coding theory and cryptography. These tutorial lecture notes provide a quick introduction to the theory of algebraic function fields. The focus is on finite constant fields since this is the only case of interest for applications to information theory. The subject is approached from the viewpoint of valuation theory. The main topics covered in these lecture notes are valued fields, valuations of algebraic function fields, divisors, the Riemann-Roch theorem, the zeta function of an algebraic function field, and the Hasse-Weil bound.