Explicitly solvable algebraic equations of degree 8 and 9

Abstract

The generic monic polynomial of degree N features N a priori arbitrary coefficients cm and N zeros zn . In this paper we limit consideration to N = 8 and N = 9 . We show that if the N —a priori arbitrary—coefficients cm of these polynomials are appropriately defined—as it were, a posteriori—in terms of 6 arbitrary parameters, then the N roots of these polynomials can be explicitly computed in terms of radicals of these 6 parameters. We also report the constraints on the N coefficients cm implied by the fact that they are so defined in terms of 6 arbitrary parameters; as well as the explicit determination of these 6 parameters in terms of the N coefficients cm .