= Lubin–Tate Galois action on primitive torsion
{c}
{title2=$\operatorname{Gal}(K_{\pi,n}/K)\simeq(\mathcal O_K/\pi^n)^{\times}$}
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 $a_\sigma\pmod{\pi^n}$. 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.
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