Function fields provide an algebraic tool for studying curves. There is a natural way in which one can define zeroes and poles for rational functions on an algebraic curve, and with appropriate constructions, there are analogous notions for function fields. Some questions which emerge immediately are:
- If we prescribe the local behaviour we want at each point, will there be a global function which has matching local behaviour at each point?
- If we place constraints on the zeroes and poles of a function, how many functions will satisfy these constraints?
These questions will form the framework for what I will cover. Firstly, however, I intend to demonstrate the analogy between algebraic curves and function fields. Then I will cover the algebraic construction of the genus (which partially
answers the rst question) and how this helps us to answer the second question using the Riemann-Roch Theorem.