# Chapter 18: Software
# 
# 
# This picture of the pencil of conics was drawn by a package called
# "quartics". It also can compute the Galois group of a quartic. First
# you must load the package:
# 
> read quartics:
# 
# A function CubicResolvent is defined which computes the cubic
# resolvent of a quartic, and then applies the criteria in the table
# above to determine the Galois group (though it cannot distinguish
# between D4 and C4). For example, let
# 
> f := x^4 - 10*x^2 + 1;

                               4       2
                         f := x  - 10 x  + 1

> CubicResolvent(f);

                                        3       2
               The cubic resolvent is, x  - 20 x  + 96 x


                     The discriminant is, 147456


             The square root of the discriminant is , 384


                     The cubic resolvent factors.


                      So the Galois Group is V.

# 
# If the quartic has real roots, the function QuarticPlot will draw the
# associated pencil of conics.
# 
> QuarticPlot(f);

> 
# 
# Maple has built-in function galois which is much more sophisticated,
# and which can compute the Galois groups
# of some polynomials of higher degree as well.