Inductive methods for counting number fields

Algebra and Number Theory Seminar
Event time: 
Tuesday, November 27, 2018 - 4:15pm
LOM 205
Robert Lemke-Oliver
Speaker affiliation: 
Tufts University
Event description: 

The motivating question of this talk is, how many number fields are there of a given flavor (e.g., with specified degree and Galois type) and bounded discriminant?  Tautologically, fields may be grouped into two classes: those which admit interesting subfields and those that don’t.  For example, degree $n$ fields whose Galois closure has Galois group $S_n$ do not admit non-trivial subfields, and such fields have been counted for $n \leq 5$.  In this talk, we instead focus on fields that do admit interesting subfields, and we propose a general framework to attack the associated counting problems.  This approach also permits one to nontrivially bound the average size of the $\ell$-torsion in the class groups of such fields, for example obtaining such results for the class groups of $D_4$ quartic fields.  This is joint work with Jiuya Wang and Melanie Matchett Wood.

Research Area(s):