Lagrange resolvent eigenvector for a cyclic field automorphism
= Lagrange resolvent eigenvector for a cyclic field automorphism
If $\sigma$ has order $n$ and the fixed field contains a primitive $n$th root $\zeta$, then
$$
T_\zeta(\beta)=\sum_{j=0}^{n-1}\zeta^{-j}\sigma^j(\beta)
$$
satisfies $\sigma(T_\zeta(\beta))=\zeta T_\zeta(\beta)$. Independence of the powers of $\sigma$ ensures that $T_\zeta(\beta)$ is nonzero for some $\beta$.