Karim BELABAS on Fri, 1 Sep 2000 18:54:14 +0200 (MET DST)

 Re: splitting field of a polynomial

```[Mathias Lederer:]
> I would like to do some calculations in the splitting field of a
> polynomial, so I thought pari/gp would be the right software for this.
>
> I imagined that there would be a function that gives a Q-basis of the
> splitting field once the polynomial is given. But I've been searching
> hard and I haven't found anything but nf and so on, which does not
> perform what I want: It gives only Q[x], where x is a zero of the
> irreducible) polynomial, and not Q[x1,...,xn], where x1,...,xn are all
> the zeros of the polynomial.

You can take the compositum of the polynomial with itself n times.

(18:49) gp > p = x^4+x+1 \\ S_4
%1 = x^4 + x + 1
(18:49) gp > polcompositum(%,p)[2]
\\ first entry would be the trivial compositum, second is Q[x1,x2]
%2 = x^12 - 27*x^9 + 67*x^8 + 475*x^6 - 1782*x^5 - 509*x^4 - 2241*x^3 +
%21083*x^2 - 42075*x + 47881
(18:49) gp > polredabs(%) \\ find a simpler defining polynomial
%3 = x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 +
%28*x^4 - 19*x^3 + 10*x^2 - 4*x + 1
(18:49) gp > polcompositum(%,p)[3] \\ Q[x1,x2,x3] = splitting field
%4 = x^24 + 8*x^23 + 36*x^22 + 124*x^21 + 355*x^20 + 870*x^19 + 1798*x^18 +
3109*x^17 + 4461*x^16 + 4592*x^15 + 174*x^14 - 13214*x^13 - 33247*x^12 -
42203*x^11 - 18156*x^10 + 32054*x^9 + 64834*x^8 + 46567*x^7 + 3587*x^6 -
16520*x^5 - 11459*x^4 - 4186*x^3 + 1599*x^2 + 4559*x + 2209
(18:49) gp > polredabs(%) \\ a nicer one
%5 = x^24 - 3*x^23 + 2*x^22 - x^20 + 4*x^19 + 32*x^18 - 35*x^17 - 92*x^16 +
%49*x^15 + 163*x^14 - 15*x^13 - 194*x^12 - 15*x^11 + 163*x^10 + 49*x^9 -
%92*x^8 - 35*x^7 + 32*x^6 + 4*x^5 - x^4 + 2*x^2 - 3*x + 1

If your example is more ambitious and the discriminant grows too large, you
may need to add a few pseudo-primes divisors to the prime table before
attempting polredabs. Checking progress at \g4 (at least) may help.

Karim.
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Karim Belabas                    email: Karim.Belabas@math.u-psud.fr
Dep. de Mathematiques, Bat. 425
Universite Paris-Sud             Tel: (00 33) 1 69 15 57 48
F-91405 Orsay (France)           Fax: (00 33) 1 69 15 60 19
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