The start of the academic year has made it much more difficult for me to get in serious posts as of late, and the number theory series has slowed.  Things should clear up at least somewhat in a few more weeks.   In the meantime, I’ll do something that occurred to me a while back but I then forgot about: posting a talk.

I took an independent study course last semester on class field theory.  As is traditional, I gave a talk last May after the course on some aspects of the subject matter.  Several faculty members at the university and teachers in my school attended, along with some undergraduates there.  In the talk, I gave an elementary overview of the p-adic numbers, assuming no more than basic number theory and point-set topology.

Anyway, I am posting the (slightly corrected) presentation and the notes here.

So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field ${F}$ with an absolute value ${\left \lvert . \right \rvert}$ like the rationals with the usual absolute value.
Definition 1 The completion ${\hat{F}}$ of ${F}$ is defined as the set of equivalence classes of Cauchy sequences:  (more…)