The monodromic Galois groups of the sextic equation of anisotropic elasticity

Abstract

It has previously been shown that the conventional algebraic Galois group of the sextic equation of anisotropic elasticity for cubic crystals is the symmetric group and the equation is therefore algebraically unsolvable in radicals. As an equation with four parameters it has also 15 monodromic Galois groups corresponding to different, relaxed, meanings of solvability in radicals. Three of these are appropriate, to solve explicitly for the functional dependence of the roots on the two directional parameters or on the two elastic parameters or on all four parameters. From the definition of a monodromic group as the root permutations induced by all complex circuits of the relevant parameters, it is shown numerically that these three monodromic groups must be either the alternating or the symmetric. The equation is therefore also unsolvable for these weaker and more appropriate meanings of solvability.