Artin Reciprocity

I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define $latex {L}&fg=000000$-series for non-abelian extensions. But for them to agree with the $latex {L}&fg=000000$-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I… Continue reading Artin Reciprocity

Some Notes on Valuations

There are some notes on valuations from the first lecture of Math 223a at Harvard. 1. Valuations Let $latex {k}&fg=000000$ be a field. Definition 1 A valuation $latex \displaystyle \left\lvert - \right\rvert : k \rightarrow \mathbb R_{\ge 0} &fg=000000$ is a function obeying the axioms $latex {\left\lvert \alpha \right\rvert = 0 \iff \alpha = 0}&fg=000000$.… Continue reading Some Notes on Valuations