I was initially planning on doing a post on Hensel’s lemma. Actually, I think I’ll leave that for later, after I’ve covered some more number theory (which may motivate it better).

So the goal for the next several posts is to cover some algebraic number theory, eventually leading into class field theory. At least in the near future, I intend to keep everything purely local. Thus, the appropriate place to start is to discuss discrete valuation rings rather than Dedekind domains. 

Absolute Values  

Actually, it is perhaps more logical to introduce discrete valuations as a special case of absolute values, which in turn generalize the standard absolute value on {\mathbb{R}}.  (more…)