Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 4 / b / Solution

Put ${\pi }_n={\zeta }_{p^n}-1$. The shifted cyclotomic polynomialis Eisenstein at $p$. Therefore it is irreducible, ${\pi }_n$ is a uniformizer, andso the extension is totally ramified. It is the splitting field of ${\Phi }_{p^n}$, and every automorphism is uniquelyThis proves the cyclotomic extension of a p-adic field isomorphismRestriction in the cyclotomic tower corresponds to reduction of $a$, so taking the inverse limit gives