I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and help me answer the question quite thourghly?
Serre's Local Fields, question 4 Chapter 2.2:
Let K be a field complete under the discrete valuation v, and let $\Omega$ be an algebraic closure of K. a)Let S be the set of subextensions E of $\Omega$ with the property that for every finite extension E' of E, e(E'/K)=1. Show that S has maximal elements. If $K_0$ is maximal, show that v prolongs to a discrete valuation of $K_0$, and that the residue field of $K_0$ is the algebraic closure of that of K.
The hint is to use: Proposition: If A is a discrete valuation ring, and if $\bar{f}$ is irreducible, then $B_f$ is a discrete valuation ring with maximal ideal $mB_f$ and residue field $K[X]/(\bar{f})$