Lubin–Tate torsion (${\mu }_{f,n}$)

The ${\pi }^n$-torsion of a Lubin–Tate formal group isIt is a free rank-one module over ${\mathcal{O}}_K/{\pi }^n{\mathcal{O}}_K$.

The two Lubin–Tate series $(1+X{)}^p-1$ and $X^p+pX$ for ${\mathbb{Q}}_p$ have respective nonzero $p$-torsion points ${\zeta }_p-1$ and the roots of $X^{p-1}=-p$. A Lubin–Tate change of series identifies their torsion fields, so