An infinite Galois group is reconstructed from the finite Galois groups of its finite subextensions. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.
Objects and notation
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.
Push the symbols
The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.
Structural reading
The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.
A hypothesis worth keeping
Subgroups in infinite Galois theory correspond to intermediate fields only after taking closure. Ramification filtrations also depend on the chosen valuation.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.