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

Let $I(L/K)$ be the inertia group and let $\phi$ denote Frobenius on the residue-field extension. The Relative Weil group isIts Weil-group topology makes $I(L/K)$ an open profinite group with its usual topology and gives $W(L/K)/I(L/K)$ the discrete topology. Thus every inertia coset is an open copy of $I(L/K)$.

Take $L=K^{\mathrm{nr}}$, the maximal unramified extension of $K$. ThenThe subgroup $\{0\}\subset \mathbb{Z}$ is open in the discrete Weil-group topology, but it is not open in the profinite topology inherited from $\hat{\mathbb{Z}}$.

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

The cyclotomic extension of a p-adic fieldhas degree $p-1$, is Galois with group $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$, and is totally ramified. Indeed, ${\zeta }_p-1$ is a root of the Eisenstein polynomial

Put $\alpha =\sqrt[p-1]{-p}$. Its polynomial $X^{p-1}+p$ is Eisenstein, so $L_2={\mathbb{Q}}_p(\alpha )$ is totally ramified of degree $p-1$. Every $(p-1)$st root of unity lies in ${\mathbb{Q}}_p$ by the Teichmuller lifts, so every root $\omega \alpha$ of this polynomial lies in $L_2$. Hence $L_2/{\mathbb{Q}}_p$ is also Galois.

For either $L_i/{\mathbb{Q}}_p$, the norm has every possible valuation because the residue-field degree is one. The norm units in a tamely totally ramified extension lie in the principal units $1+p{\mathbb{Z}}_p$: reduction of a unit norm is the $(p-1)$st power of its residue, hence is $1$. Part (b) says that this unit norm subgroup has index $p-1$, exactly the index of $1+p{\mathbb{Z}}_p$ in ${\mathbb{Z}}_p^{\times }$. ConsequentlyThe uniqueness clause in the existence theorem of local class field theory now gives