Thanks for that page, Bernhard. I find it useful for quadratic fields with class number h > 1. The point of starting this thread, was to investigate the determinants, leading coefficients and constants for higher degree polynomials, such as 200degree or higher.
The discriminant of the cubic polynomial a*x^3 + b*x^2 + c*x + d is b^2*c^2  4*a*c^3  4*b^3*d  27*a^2*d^2 + 18*a*b*c*d.
I tried one example with P(x) = X^3 + X + 9 with discriminant d =
0^2*1^2  4*1*1^3  4*9^3*0  27*1^2*9^2 + 18*1*0*1*9 = 2191 = 7*313
Using the p*q constant product I was talking about earlier, I solved for Q(x) discriminant = 2191.
Q(x) = 3*X^3 + b*X^2 + c*X + 3
2191 = b^2*c^2  4*3*c^3  4*b^3*3  27*3^2*3^2 + 18*3*b*c*3
2191 = b^2*c^2  12*c^3  12*b^3  2187 + 162*b*c
I wasn't able to find any straightup solutions unfortunately  any help?
