= Lubin–Tate extension
{c}
{title2=$K_{\pi,n}=K(\mathcal F[\pi^n])$}
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.
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