Time to go back to basic algebraic number theory (which we’ll need for two of my future aims here: class field theory and modular representation theory), and to throw in a few more facts about absolute values and completions—as we’ll see, extensions in the complete case are always unique, so this simplifies dealing with things like ramification. Since ramification isn’t affected by completion, we can often reduce to the complete case. 

Absolute Values  

Henceforth, all absolute values are nontrivial—we don’t really care about the absolute value that takes the value one everywhere except at zero.

I mentioned a while back that absolute values on fields determine a topology. As it turns out, there is essentially a converse. 

Theorem 1 Let {\left|\cdot\right|_1}, {\left|\cdot\right|_2} be absolute values on {K} inducing the same topology. Then {\left|\cdot\right|_2} is a power of {\left|\cdot\right|_1}  (more…)