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Old 2017-12-26, 21:30   #3
carpetpool
 
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"Sam"
Nov 2016

5148 Posts
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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 200-degree 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 straight-up solutions unfortunately --- any help?
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