# 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.