Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 28 / 4 / a / Solution

Expand the translated cyclotomic polynomial:Every nonleading coefficient is divisible by $p$, and the constant term is not divisible by $p^2$. The polynomial is Eisenstein, hence irreducible. Its root $\lambda ={\zeta }_p-1$ is a uniformizer of $E={\mathbb{Q}}_p({\zeta }_p)$, and the Eisenstein polynomial valuation criterion givesThus this local cyclotomic extension is totally ramified.