Proof of the local existence theorem

Let K be a nonarchimedean local field. As explained in the previous lecture, to complete our proof the existence theorem of local class field theory (Theorem 27.8), which states that every finite index open subgroup of K× is a norm group, we need to construct a system of totally ramified abelian extensions Kπ,n/K and a homomorphism θπ : K × → Gal(Kab π /K) satisfying the hypotheses of Theorem 32.1. More precisely, let p be the maximal ideal of the valuation ring A := OK , let π be a uniformizer for p, and define: • q := A/p is the cardinality of the residue field; • Km/K is the unique extension of degree m in Kunr; • Kπ,n,m := Kπ,nKm; • Uπ,n,m := (1 + pn)〈πm〉 ⊆ K×; • Kπ := ⋃ n≥1Kπ,n; • Kab π := KπK;