**
Peter Doyle
and
Curt McMullen
**

**
Last revised 1989
Version 1.0A1 dated 15 September 1994
**

Equations that can be solved using iterated rational maps are characterized: an equation is `computable' if and only if its Galois group is within 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 Kronecker) 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.

- Introduction.
- Galois Theory of Rigid Correspondences.
- Purely Iterative Algorithms.
- Towers of Algorithms.
- Rational Maps with Symmetry.
- References