Solving the quintic by iteration

Equations that can be solved using iterated rational maps are character-ized: an equation is ‘computable ’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermite and Kro-necker) is replaced by a purely iterative algorithm. The algorithm requires a rational map with icosahedral symmetries; we show all rational maps with given symmetries can be described using the classical theory of invariant polynomials. 1 Introduction. According to Dickson, Euler believed every algebraic equation was solvable by radicals [2]. The quadratic formula was know to the Babylonians; solutions of cubic and quartic polynomials by radicals were given by Scipione del Ferro, Tartaglia, Cardano and Ferrari in the mid-1500s. Abel’s proof of the insolvability of the general quintic