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

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.