One Substitution, Four Degrees, and the Number That Stops Them

Abstract

Textbooks present the solution of algebraic equations as four unrelated stories. We show that a single substitution, x = w - c/w with c = p/n, together with a single target, the Dickson polynomial equation, accounts for all of them. Bringing an arbitrary equation of degree n to the target form costs an auxiliary equation of degree (n-1)!/2, the number of cyclic orderings of the roots. The mechanism solves the equation outright exactly while (n-1)!/2 < n, that is, up to degree four. Used instead as a test for solvability by radicals, it is exact if and only if n is prime, because the normalizer of the dihedral group in the symmetric group has order n times Euler’s totient of n, which equals the order of AGL(1,n) precisely then. We also determine the degree of the minimal factor of the auxiliary equation and the field it generates, and verify everything on explicit equations of degrees 5, 7, 11, and 13.