Resolvent (Galois theory)

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 roots ${\alpha }_1,{\alpha }_2,{\alpha }_3$ of a depressed cubic $X^3+aX+b$ over a field of characteristic zero, choose a primitive cube root of unity $\omega$. Its Lagrange resolvent for a cubic consists of the Fourier combinationsCyclically permuting the roots multiplies these combinations by cube roots of unity, so their cubes are fixed by the cyclic subgroup. Direct expansion gives $uv=-3a$ and $u^3+v^3=-27b$. Their cubes are therefore the roots of the cubic resolvent $Y^2+27bY-27a^3$.

Since the roots sum to zero, ${\alpha }_1=(u+v)/3$. The quadratic equation for the cubes, followed by compatible cube roots, gives the mechanism of the Cardano formula. A primitive root is required: choosing $\omega =1$ would make both combinations vanish.

For $x^3+ax+b$, the Lagrange-resolvent cubes are roots of $X^2+27bX-27a^3$.

For a depressed cubic $X^3+aX+b=0$ over a field of characteristic zero, putChoose $U,V$ with $U^3=-b/2+\sqrt{D}$, $V^3=-b/2-\sqrt{D}$, and $UV=-a/3$. Then $x=U+V$ is a root, becauseFor $a\ne 0$, choose a nonzero $U$ and set $V=-a/(3U)$; this enforces the compatibility rather than choosing two arbitrary cube roots. If $a=0$, take a cube root of $-b$ directly. With a primitive cube root of unity $\omega$, the other roots are $\omega U+{\omega }^{2}V$ and ${\omega }^{2}U+\omega V$.

The Lagrange resolvent for a cubic explains why a quadratic equation appears inside the formula. Even when the cubic has three real roots, complex intermediate radicals may occur; the compatible pairing of cube roots is what makes the resulting expressions solve the original equation.