Student Arithmetic Seminar

It is known that ray class fields over Q are the cyclotomic extensions. Over imaginary quadratic fields, the ray class fields are constructed "analytically" using the theory of complex multiplication. But the computations arising from this method are elaborate and sometimes need a computer even in the simplest of cases. We will instead use Kummer theory to compute examples of class fields and explain why it suffices, in principle.

We will assume familiarity with Kummer theory. We will compute with Kummer extensions of cyclotomic extensions of Q (and Qp); so some familiarity with explicit computations may be helpful. I will however explain notions from class field theory that are relevant. Speaker(s): Kannappan Sampath (University of Michigan)