Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 136 / 3 / c / Solution

Over $K={\mathbb{Q}}_p$, consider the two Lubin–Tate seriesBoth have linear term $pX$ and reduce to $X^p$ modulo $p$. The nonzero roots of $f$ are ${\zeta }_p^i-1$, while the nonzero roots of $g$ satisfy

The Lubin–Tate change of series $h:{\mathcal{F}}_f\to {\mathcal{F}}_g$ and its inverse have coefficients in ${\mathbb{Z}}_p$ and converge on the maximal ideal. They therefore give mutually inverse bijections between the first torsion sets without changing the fields generated by them. Hence the first Lubin–Tate torsion fields for the p-adic numbers satisfySince ${\mathbb{Q}}_p({\zeta }_p-1)={\mathbb{Q}}_p({\zeta }_p)$, the required equality follows.