An extension valuation rescales the base valuation by the ramification index. The point is to make the formal expression readable enough to audit line by line.
Start locally
Infinite Galois groups carry the Krull topology and become profinite groups. For a discretely valued field \(K\), completions and residue fields add a second layer of arithmetic to extensions \(L/K\).
The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.
Compute before generalising
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
The global view
What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.
Edge conditions
Subgroups in infinite Galois theory correspond to intermediate fields only after taking closure. Ramification filtrations also depend on the chosen valuation.
With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.