I've been looking at Janusz book Algebraic Number Fields. I can't seem to figure out the following correspondence:
Let $\mathfrak{p}$ be a finite prime of a field $K$ and $L/K$ a finite Galois extension. Then we get a valuation ring $R$ from $\mathfrak{p}$ by picking one valuation. Janusz then writes (p. 123) "Identify $\mathfrak{p}$ as the maximal ideal of $R$". The problem is that given the maximal ideal in the valuation ring $R$, I can't see how we get back a valuation on $K$ unless we assume that the ring $R$ is a DVR?
I must be missing something here? Finally if we then let $R'$ be the integral closure of $R$ in $L$ and $\mathfrak{P}$ some prime containing the maximal ideal of $R$ should this somehow define a prime on $L$?