A Structural Classification of Solvable Sextics: From Triple Sum Resolvents to Five Collapse Families

This paper presents a complete classification of solvable sextic equations by radicals, building upon a structural resolvent framework derived from the representation theory of the symmetric group S_6. We begin by reducing the general sextic to Joubert canonical form x^6 + α x^4 + β x^2 + γ x + γ = 0. Using the induced permutation representation on the 20 triple sums of roots, we decompose the representation space into irreducible components of dimension 5, yielding a canonical quintic resolvent Φ(y). We prove that the constant term of this resolvent is c_0 = K(4α^3 - β^2 - 4γ^2), where K is a non-zero normalization constant. When c_0 = 0, i.e., 4α^3 - β^2 - 4γ^2 = 0, the resolvent collapses and the sextic becomes solvable by radicals. We identify and classify five distinct families under this condition: · Family I: Degenerate sextics factoring as cubic × cubic · Family II: Even sextics with γ = 0 · Family III: Resolvent sextics with α = -3,\ γ = 16 + 4β · Family IV: Sextics admitting quartic–quadratic factorization · Family V: A new family defined by 27γ^2 = (4α - 3β)^3, with generic Galois group S_3 × C_2 For each family, we provide explicit algebraic conditions, complete derivations, Galois-theoretic interpretations, structural analysis of collapse and branch points, and worked examples with exact radical expressions. The independence of the fifth family is rigorously analyzed through intersection theory of algebraic varieties. This work unifies classical resolvent theory with modern representation-theoretic methods and provides a ready-to-use classification for researchers in Galois theory, algebraic geometry, and symbolic computation.