Lubin–Tate Galois action on primitive torsion ($\mathrm{Gal}(K_{\pi ,n}/K)\simeq ({\mathcal{O}}_K/{\pi }^n{)}^{\times }$)

For a primitive Lubin–Tate torsion point ${\lambda }_n$, every Galois automorphism has $\sigma ({\lambda }_n)=[a_{\sigma }]({\lambda }_n)$ for a unique unit class $a_{\sigma }(\mathrm{mod}{\pi }^n)$. All torsion points lie in $K({\lambda }_n)$ because the integral endomorphism series converge there; hence the torsion field is Galois. The displayed map is an injective homomorphism. Its domain has order $(q-1)q^{n-1}$ by the primitive Eisenstein polynomial, equal to the number of unit classes, so it is an isomorphism.