= Uniform unramified division field over a local field
{title2=$[L:K]\leq n[K(E[n]):K]$}
Let $E/K$ have <good reduction> over a <local field> of residue characteristic $p$, with $p\nmid n$. All solutions of $nQ=P$, for all $P\in E(K)$ together, lie in a single finite <unramified extension> of degree at most $nf$, where $f=[K(E[n]):K]$. First lift a solution over the algebraic closure of the <residue field> to a finite <unramified extension>; the <multiplication isomorphism of a formal group law> uniquely corrects the error in $E_1$. This also puts $E[n]$ in the <maximal unramified extension>. If $\sigma$ is its Frobenius generator, $\sigma^f$ fixes $E[n]$. For $nQ=P$, set $T=\sigma^fQ-Q\in E[n]$. Then $\sigma^{nf}Q=Q+nT=Q$, uniformly in $P$.
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