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Old 2017-02-17, 02:02   #10
carpetpool
 
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"Sam"
Nov 2016

2·3·53 Posts
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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
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