The structure of the class group in global function fields

The parallelism between algebraic number fields and global function fields is both beautiful and useful. Frequently the analogy is used to provide insight into the number field realm by formulating suitable analogues of results previously known for function fields. The analogy is useful in the opposite direction as well. For example, less is known about the structure of the class group in the function field situation than in the number field case. Unlike many other areas of number theory, no one has been able to use the powerful methods of algebraic geometry to simplify the function field situation.; In this dissertation, we prove function field analogues of some number field results, demonstrating the existence of infinitely many function fields of any fixed degree with ideal class numbers divisible by any given positive integer greater than 1. More precisely, we show that for any integers m and n with m, n > 1, there are infinitely many function fields K of degree m over the rational function field such that the ideal class group of K contains a subgroup isomorphic to ($Z/nZ$)m−1 if the prime at infinity is totally ramified in K and $Z/nZ$ if the prime at infinity splits completely in K. In the special case that m is a prime dividing q − 1, we can construct a lower bound on the number of cyclic function fields of degree m with class number divisible by n.