[GAP Forum] help with solving equations on finite fields..

[Hulpke,Alexander]

Dear Forum, Dear Maycon Carneiro,

> I creat an AlgebraicExtension over GF(2) using the polynomial x^3+x^2+1,
> 
> gap>z2:=GF(2);
> x:=X(z2,"x");
> pol:=x^3+x^2+1;
> F8:=AlgebraicExtension(z2,pol);
> a:=RootOfDefiningPolynomial(F8);
> 
> now, i'd like to get the solutions over F8 of the equation:
> 
> z^2+z*y+a*y^2= c, where c is an especified element of F8, say a^2.. and z,
> y are variables in F8
> 
> how can i do this? or at least in one variable, say z, and i'll put an
> especified element for y..

There is no general polynomial equations solver. in GAP.
Basically you’ll need to do the same as in the answer I already gave you in

https://math.stackexchange.com/questions/2563103/how-to-solve-a-quadratic-equation-over-finite-fields-with-gap

Note that by using `Algebraic Extension’ your calculation will be a magnitude slower than if you use the built-in GF(8). I’ve also shown you in

https://math.stackexchange.com/questions/2565387/how-to-work-in-gap-with-finite-fields-given-by-a-particular-polynomial

how you can represent elements of (in this case) GF(8) in terms of roots of your original polynomial.

Regards,

    Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke