Lagrange resolvent

Summary

A Lagrange resolvent is a sum $\alpha + \zeta \sigma(\alpha) + \cdots + \zeta^{n-1} \sigma^{n-1}(\alpha)$ built from an element $\alpha$, an automorphism $\sigma$ of order $n$ and an $n$-th root of unity $\zeta$. The automorphism $\sigma$ multiplies it by $\zeta^{-1}$, so its $n$-th power is fixed by $\sigma$. This turns a cyclic extension into an extension obtained by adjoining an $n$-th root, which is the key step in solving equations by radicals. Lagrange used such sums of the roots to explain the formulas for cubic and quartic equations.