View Single Post
Old 2017-02-17, 16:19   #14
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

104016 Posts
Default

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).

The same number field, you say? I'll ask Pari for a cursory description:

? ?polsubcyclo
polsubcyclo(n,d,{v=x}): finds an equation (in variable v) for the d-th degree subfields of Q(zeta_n). Output is a polynomial or a vector of polynomials is there are several such
fields, or none.[/I]

The d-th degree subfields are not the same as the whole cyclotomic field if d < phi(n).

(17:57) gp > polsubcyclo(2201, 5)
[...]
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


"The same number field as the cyclotomic polynomial d" you say? Eh? And it gives polynomials for all subfields of degree d, whether there are any or not (if there are none, it gives an empty vector).

I also don't know what you mean, "the same number field properties." Do you?

EDIT:

The mistake is the polynomials are degree 5, not 4!


That's not the only mistake. x^5 + x^4 + x^3 + x^2 + x + 1 is not irreducible. It is the product of the cyclotomic polynomials for the primitive 2nd, 3rd, and 6th roots of unity. The splitting field is the field Q(zeta_3), which is of degree 2.

Let's see here, 2201 factors as 31*71. Q(zeta_2201) contains Q(zeta_31) and Q(zeta_71), so also the degree-5 subfield of Q(zeta_31), and the degree-5 subfield of Q(zeta_71). These will have discriminants of 31^4 and 71^4, respectively. There are also 4 other degree-5 subfields, whose discriminants are divisible by both 31 and 71 (probably all 31^4 * 71^4, but I'm too lazy to check).

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. You might, after all, specify things beyond the realm of possibility. Besides, solving high-degree multivariate polynomial equations is, at best, time consuming even if there happen to be solutions.

I would suggest that you learn the basics of the "theory of equations" before proceeding further.

Last fiddled with by Dr Sardonicus on 2017-02-17 at 16:21
Dr Sardonicus is offline   Reply With Quote