Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 27 / 1 / b / Solution

The shifted cyclotomic polynomial ${\Phi }_{p^{n+1}}(1+T)$ is Eisenstein at $p$: its constant coefficient is $p$, and modulo $p$ it is $T^{p^n(p-1)}$. Thus $[K_n:{\mathbb{Q}}_p]=p^n(p-1)$, ${\zeta }_n-1$ is a uniformizer, and $[K_n:K_{n-1}]=p$ for $n\ge 1$. The $p$ automorphisms over $K_{n-1}$ send ${\zeta }_n$ to $\xi {\zeta }_n$ for $\xi \in {\mu }_p$.

Since ${\zeta }_n-1$ has positive valuation, evaluating an integral formal power series there converges. A unit series has unit constant coefficient and evaluates to a unit. Evaluate the Coleman norm operator identity at $T={\zeta }_n-1$:The final equality uses norm fixation and the compatible choice of roots. HenceIn particular, norm fixation of a single Coleman power series produces a norm-compatible sequence of local units.