Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 136 / 5

Restriction to the maximal unramified extension gives a surjectionThe abelian Weil group is the inverse image of the dense subgroup $\mathbb{Z}\subset \hat{\mathbb{Z}}$ generated by Frobenius:It contains the full inertia kernel. Since $\mathbb{Z}$ is dense in $\hat{\mathbb{Z}}$, the inverse image is dense in $\mathrm{Gal}({\mathbb{Q}}_p^{\mathrm{ab}}/{\mathbb{Q}}_p)$ with its profinite topology.

Solved by gpt-5.6-sol high.

The translated cyclotomic polynomialis Eisenstein at $p$. Hence ${\zeta }_{p^n}-1$ generates a totally ramified extension of degreeEvery automorphism sends ${\zeta }_{p^n}$ to ${\zeta }_{p^n}^a$ for a unique $a\in (\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$. This gives an injectionand equality of orders makes it an isomorphism.

Normalize the Local Artin map so that $p$ maps to arithmetic Frobenius on the maximal unramified extension. For $u\in {\mathbb{Z}}_p^{\times }$, define its action on $K_n$ bywhere using $u$ rather than $u^{-1}$ gives the opposite Frobenius convention. These compatible maps, together with the image of $p$, define reciprocity on $p^{\mathbb{Z}}\times {\mathbb{Z}}_p^{\times }={\mathbb{Q}}_p^{\times }$ and hence on every finite abelian extension by the Local Kronecker-Weber theorem.

Solved by gpt-5.6-sol high.

Local Artin reciprocity states that the continuous maphas dense image and induces, for every finite abelian extension $L/K$, an isomorphismThus the norm subgroup of a local field extension is the kernel of the Artin map restricted to $L$.

For $L=K_n$, the uniformizer $p$ maps to the unramified Frobenius and therefore acts trivially on the totally ramified cyclotomic extension. By part b, a unit $u$ acts trivially on $K_n$ exactly when $u\equiv 1(\mathrm{mod}{p}^n)$. Sincethe kernel, and hence the norm subgroup, is

Solved by gpt-5.6-sol high.