UC San Diego

In this dissertation, we undertake the study of the class numbers of fields of bounded relative degree. Fix $B > 1$ and let $\mathcal}B}(B)$ be the set of all number fields $M$ such that $M$ can be reached by a tower of fields, $\Q = M_0 \subset M_1 \subset M_2 \subset \cdots \ subset M_n = M$ such that $[M_i:M_}i-1}] \leq B$ for $1 \ leq i \leq n$. Building on the work of Harold Stark and Andrew Odlyzko, we show that there for a fixed $B$ there are only finitely many CM fields $M$ of degree greater than or equal to 387 with a given class number. In the process of proving this, we also obtain lower bounds for the residue of Dedekind zeta functions and $L(1, \chi)$. We also obtain some upper bounds for these functions by mimicking some of Jeffrey Hoffstein's calculations [Hof79]