With the school year starting, I can’t keep up with the one-post-a-day frequency anymore. Still, I want to keep plowing ahead towards class field theory.

Today’s main goal is to show that under certain conditions, we can always extend valuations to bigger fields. I’m not aiming for maximum generality here though.

Dedekind Domains and Extensions

One of the reasons Dedekind domains are so important is

Theorem 1 Let ${A}$ be a Dedekind domain with quotient field ${K}$, ${L}$ a finite separable extension of ${K}$, and ${B}$ the integral closure of ${A}$ in ${L}$. Then ${B}$ is Dedekind. (more…)