 Lubin–Tate extension

Fix a uniformizer $\pi$ of a local field $K$ and a Lubin–Tate formal group $\mathcal{F}$. Its nth torsion field is generated by all its ${\pi }^n$-torsion points. A primitive point ${\lambda }_n$ generates the Lubin–Tate torsion as a free rank-one ${\mathcal{O}}_K/{\pi }^n$-module; therefore $K_{\pi ,n}=K({\lambda }_n)$. If the residue field has size $q$, the Eisenstein layers of Lubin–Tate torsion give $[K_{\pi ,n}:K]=(q-1)q^{n-1}$ and total ramification.