For odd \(p\), \({\operatorname{Tr}}_{E/L}(\alpha)=0\). Choose \(b\) with \(qb\equiv-1\pmod p\), let \(a=(qb+1)/p\), and set \(z=t_L^a\alpha^b\). Then \(z\) is a uniformizer.
For its Eisenstein polynomial \(f\) and constant-coordinate projection \(P_0\), the normalized corestriction uses \(\kappa=-f'(z)/z^{p-1}\). Writing \(u_0=s(P_0)\),
\[
\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\qquad
{\operatorname{tr}}_L\operatorname{Lift}_E(C)
={\operatorname{Tr}}_{E/L}(\kappa^{-1}{\operatorname{tr}}_EC).
\]
If \({\operatorname{tr}}_EC=0\), the lift has trace zero. Adding \(\alpha I_m\) preserves trace zero. Induction therefore gives trace zero at every native stage and every native correction.
Reading the diagram
This is stronger than the automatic ambient trace cancellation from a multiplicity divisible by \(p\). Native trace controls the scalar restriction used in full-character construction.