Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 24 / 3 / ii

For a principal unit $u$, apply the Hensel lemma to $F(T)=T^{p-1}-u$. The residue class one is a root, and $F^{\prime }(1)=p-1$ is a unit. Thus there is a unique $v\in 1+{\pi }_K{\mathcal{O}}_K$ withIn particular it is a unit of ${\mathcal{O}}_K$. Choose this $v$ for the $u$ just obtained and put $\alpha ={\pi }_K/v$. Then ${\alpha }^{p-1}=-p$, so ${\mathbb{Q}}_p(\alpha )\subseteq K$. The polynomial $T^{p-1}+p$ is Eisenstein, giving degree $p-1$ for the left-hand field. The p-adic cyclotomic extension $K$ also has degree $p-1$, so