Construction of polynomials with same discriminants
Hi all,
For the monic polynomial P(x) = x^n+a_1*x^(n1)+a_2*x^(n2)+......+a_(n2)*x^2+a_(n1)*x+r where r = p*q, there exists a polynomial Q(x) = p*x^n+a_1*x^(n1)+a_2*x^(n2)+......+a_(n1)*x^2+a_n*x+q with the same discriminant as P(x) and defines the same field as P(x). In fact, there should be a simple method to perform the construction of Q(x) from P(x) for any degree n. Does anyone know an easy method for constructing Q(x)?
For quadratics n = 2, it is easy to construct. The quadratic polynomial f = a*x^2 + b*x + c has discriminant d = b^24*a*c. Hence the polynomials c*x^2 + b*x + c and x^2 + b*x + a*c have the same discriminant as f.
For instance, let P(x) = x^2 + 9*x + 21.
P(x) has discriminant 9^24*1*21 = 3 and 21 = 3*7. The construction of Q(x) would use 3 and 7 as the leading coefficients and or ending coefficients.
Q(x) = 3*x^2 + 9*x + 7 has the same discriminant as P(x): 9^24*3*7 = 3.
What about cubic polynomials a*x^3 + b*x^2 + c*x + d?
When we are given P(x) = x^3 + a*x^2 + b*x + r where r = p*q, how can one construct Q(x) = p*x^3 + a_2*x^2 + b_2*x + q with the same discriminant as P(x) and defines the same field as P(x)?
For instance, take the cubic polynomial P(x) = x^3 + 4*x^2  x + 15 which has discriminant d = 10975. What is the polynomial Q(x) = 3*x^3 + a*x^2 + b*x + 5 (or reversed possibly) with discriminant d = 10975 and defining the same field as P(x)? More specifically, is one able to show the work for the construction of Q(x)?
Is this construction easy to do for say, 200degree polynomials or higher? (I do believe it is possible, I don't know the complexity of it however.)
Thanks for help, comments, and suggestions.
Last fiddled with by carpetpool on 20171226 at 18:36
