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.
Climbing Mount Bourbaki 
September 16, 2009 at 6:22 pm
It’s a shame combinatorialists and number theorists don’t talk more often. The construction of the ring![R[[x]]](https://s0.wp.com/latex.php?latex=R%5B%5Bx%5D%5D&bg=ffffff&fg=29303b&s=0&c=20201002)
of formal power series with coefficients in a commutative ring 
is almost identical to the construction of the 
-adic numbers (as you’ve discussed in some generality already) and combinatorialists owe all the nice convergence properties of formal power series to the properties of the 
-adic topology, which is induced by essentially the same non-Archimedean metric as in the 
-adic case. But I don’t think this fact gets recognized very often in combinatorics classes.
September 17, 2009 at 8:59 am
In the last section you put Kronecker-Weber under “Why does Q_p matter”. Is there any relation between them actually..?
September 17, 2009 at 8:59 am
Oops I overlooked the local formulation. Please delete these two comments.