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 2017-02-17, 02:02 #10 carpetpool     "Sam" Nov 2016 2·3·53 Posts Check this out sm88, (17:57) gp > polsubcyclo(2201, 5) %33 = [x^5 - x^4 - 880*x^3 + 176*x^2 + 179584*x + 26624, x^5 + x^4 - 28*x^3 + 37*x^2 + 25*x + 1, x^5 - x^4 - 880*x^3 + 6779*x^2 + 14509*x - 112039, x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5, x^5 - x^4 - 880*x^3 + 15583*x^2 - 95541*x + 196101, x^5 - x^4 - 880*x^3 - 2025*x^2 + 49725*x - 112039] I figured out the idea to this: polsubcyclo(n, d) will give ALL polynomials with the same number field as the cyclotomic polynomial d if and only if d | phi(n). I've organized the new polynomials. Dr. Sardonicus should take a look at these ones: x^5 - x^4 - 880*x^3 + 176*x^2 + 179584*x + 26624 x^5 + x^4 - 28*x^3 + 37*x^2 + 25*x + 1 x^5 - x^4 - 880*x^3 + 6779*x^2 + 14509*x - 112039, x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5 x^5 - x^4 - 880*x^3 + 15583*x^2 - 95541*x + 196101 x^5 - x^4 - 880*x^3 - 2025*x^2 + 49725*x - 112039 All 5 appear to have the same number field properties as x^5+x^4+x^3+x^2+x+1 EDIT: The mistake is the polynomials are degree 5, not 4! Command: (17:54) gp > ?? polsubcyclo() polsubcyclo(n,d,{v = 'x}): Gives polynomials (in variable v) defining the sub-Abelian extensions of degree d of the cyclotomic field Q(zeta_n), where d | phi(n). If there is exactly one such extension the output is a polynomial, else it is a vector of polynomials, possibly empty. To get a vector in all cases, use concat([], polsubcyclo(n,d)). The function galoissubcyclo allows to specify exactly which sub-Abelian extension should be computed. The library syntax is GEN polsubcyclo(long n, long d, long v = -1), where v is a variable number. Last fiddled with by carpetpool on 2017-02-17 at 02:15