Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 136 / 1 / b / Solution

The main theorem of local class field theory gives a continuous Local Artin mapwith dense image, normalized by sending a uniformizer to a chosen Frobenius. For every finite abelian extension $L/K$, it induces the Local Artin reciprocity isomorphismThe existence theorem of local class field theory says that the finite-index open subgroups of $K^{\times }$ are exactly the norm subgroups $N_{L/K}(L^{\times })$ for finite abelian extensions $L/K$, and that the extension is uniquely determined inside $K^{\mathrm{ab}}$.

Write $e=e(L/K)$ and $f=f(L/K)$. The valuation of a field norm satisfiesso the valuation image of the norm subgroup is $f\mathbb{Z}$. The exact sequence obtained from $v_K:K^{\times }\to \mathbb{Z}$ therefore givesThe left side is $[L:K]=ef$ by Local Artin reciprocity. Cancelling $f$ proves