resolvent

A resolvent is a helper polynomial, built from the roots of the one you actually care about, designed so that its own behaviour exposes hidden structure of the original. By watching whether the resolvent factors and how, you can pin down which group of symmetries the original polynomial has — a detective's tool for computing Galois groups by hand.

Concretely, given a polynomial f with roots r_1, ..., r_n and a subgroup H of S_n, one forms an auxiliary expression in the roots (a multivariate polynomial) and lets the full symmetric group permute the variables. The orbit of that expression yields the roots of a resolvent polynomial whose coefficients, being symmetric in the r_i, lie in the base field. The classical Lagrange resolvent and the Galois resolvent (a primitive element of the splitting field) are the historical sources of the term; the linear Lagrange resolvent (1/n) sum of zeta^{-k} sigma^k(alpha) is central to solving cyclic equations by radicals.

The diagnostic principle is this: the resolvent attached to a subgroup H acquires a root in the base field exactly when the Galois group of f is conjugate into H. So by computing several resolvents and checking which ones acquire rational roots (and factoring them), one narrows down the Galois group among the transitive subgroups of S_n. This is the computational backbone of algorithms that determine Galois groups of low-degree polynomials.