This program illustrates the Galois group of a quartic polynomial with coefficients in C(z). To vary the parameter z, click on the plane where you wish it to go. Do it in small steps to avoid numerical errors. Each step uses the previous step as initial data for a Newton iteration to compute the roots. As z (represented by a gray point) varies in the complex plane, the roots of P vary along with it (represented by the red, blue, green and brown points). The pink points represent the branch points and by going around them and returning to the origin we see the action of Galois. All five possible groups for an equation of degree four are illustrated. Select the group you wish to study in the last line of the box and then hit the reset button. I am also making avaliable some explanatory notes on the mathematics involved.
This program is based on a program written for a cubic by Philip D. Smolen in a class taught by J. Tate. The program was adapted by F. Voloch for quartics. Most changes were in the polynomial algebra.
You can download the Tcl/Tk source text for the example by right-clicking here. There is also a standalone Windows executable. Philip made some changes to the program to fix a problem when one goes too close to a branch point. It's not fully tested but it's available here.
Other programs.
This page is maintained by Felipe Voloch