A resolvent is an auxiliary polynomial built from expressions in the roots of a polynomial. The action of its Galois group on those expressions gives information about solvability and intermediate field extensions. A Lagrange resolvent for a cubic uses Fourier combinations of the three roots.
For cubic roots and a cube root of unity , the Fourier combinations and have cubes satisfying a quadratic resolvent.
Cardano's formula solves a depressed cubic by taking square roots in its quadratic resolvent and then compatible cube roots.
Articles by others on the same topic
In the context of Galois theory, a **resolvent** is an auxiliary polynomial that is used to study the roots of another polynomial, particularly in relation to the solvability of polynomials by radicals. The concept primarily arises within the field of algebra when investigating the solutions of polynomial equations and their symmetries.