Here is an example for quartic polynomials, using Wolfram Alpha I solved for the last three coefficients.
x^4+3x^3+4x^2+2x+1
with discriminant D = 125
has the same number field properties as the cyclotomic polynomial x^4+x^3+x^2+x+1.
Is PARI GP capable of doing exactly this for degree 6 polynomials, and finding polynomials with the same number field properties as x^6+x^5+x^4+x^3+x^2+x+1.
One way would be setting up a few queries via PARI GP to solve for:
Discriminant [ax^6+bx^5+cx^4+dx^3+ex^2+fx+g] for a = 1, 20; b = 1, 20; c = 1, 20; d = 1, 20; e = 1, 20; f = 1, 20; g = 1, 20; where 1, 20 denotes all integers 1 <= (a, b, c, d, e, f, g) <= 20.
The number of combinations of polynomials would be, 20^7 = 1280000000 such polynomials searching for discriminant = 16807 or some other related value.
Please show me a command via Pari GP which can do five variable solution for:
Discriminant of x^6+4x^53x^4+ax^3+bx^2+cx+d = 16807
There should be at least one set of integers which will work with the following property.
This is interesting, especially for getting into polynomials of degrees 6, 10, and 12.
