Completely Solving the Quintic by Iteration

AbstractIn the late nineteenth century, Felix Klein revived the problem of solving the quintic equation from the moribund state into which Galois had placed it. Klein’s approach was a mix of algebra and geometry built on the structure of the regular icosahedron. His method’s key feature is the connection between the quintic’s Galois group and the rotational symmetries of the icosahedron. Roughly a century after Klein’s work, Doyle and McMullen developed an algorithm for solving the quintic that also exploited icosahedral symmetry. Their innovation was to employ a symmetrical dynamical system in one complex variable. In effect, the dynamical behavior provides for a partial breaking of the polynomial’s symmetry and the extraction of two roots following one iterative run of the map. The recent discovery of a map whose dynamics breaks all of the quintic’s symmetry allows for five roots to emerge from a single pass. After sketching some algebraic and geometric background, the discussion works out an explicit procedure that deploys the special map in order to solve the quintic in a complete sense.