Re: How to declare a finite field element resulting from a Prime Power in Pari/ɢᴘ ?

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Fri, Jul 04, 2025 at 12:38:49PM +0200, Laël Cellier wrote:
> Let’s talk first about the equivalent SageMath code : I’ve a finite field
> declared as ff= GF(q) where q is the prime power of degree 12.
> 
> Given a finite field element that SageMath printf as |0*z12^11 + 0*z12^10 +
> 16260673061341949275257563295988632869519996389676903622179081103440260644990*z12^9
> + 0*z12^8 + 0*z12^7 + 0*z12^6 + 0*z12^5 + 0*z12^4 +
> 11559732032986387107991004021392285783925812861821192530917403151452391805634*z12^3
> + 0*z12^2 + 0*z12 + 0, how to set a variable belonging to ff to such static
> value in Pari/ɢᴘ ? 

You need to find the minimal polynomial P of z12 in SageMath and the characteristic p.

Then do in GP

z12 = ffgen(P*Mod(1,p),'z12);

Then you can do

a = 0*z12^11 + 0*z12^10 + 16260673061341949275257563295988632869519996389676903622179081103440260644990*z12^9 + 0*z12^8 + 0*z12^7 + 0*z12^6 + 0*z12^5 + 0*z12^4 + 11559732032986387107991004021392285783925812861821192530917403151452391805634*z12^3 + 0*z12^2 + 0*z12 + 0;

to define a.

> Since finite field elements in such cases are polynomes…|

No they are not, they are only displayed as a polynomial expression in the generator.

Cheers,
Bill