Quote:
Originally Posted by Dr Sardonicus
If you want polynomials defining the same field as any given one (which I assume is monic and irreducible), it's very easy to find any number of them, but the methods I know do NOT involve specifying coefficients in advance.

Is there a pdf or wiki page I could read to find these methods, or just explain or post them here? If the method you are thinking of doesn't use fixed coefficients, how does it get across finding one of infinitely many polynomials defining the same field as polynomial P?
This time, using logic, and trial and error
x^6+14x^4+21x^2+7 discriminant D = 2^6*7^5*13^4 defines the same (or very similar) field as x^6+x^5+x^4+x^3+x^2+x+1 discriminant D = 7^5.
Other methods should find more degree 6 polynomials defining the same field as the cyclotomic polynomial for 7.