# [GAP Forum] finite fields question

Alexander Hulpke hulpke at math.colostate.edu
Wed May 19 17:35:51 BST 2004

```Dear David, Dear GAP-forum:

> Possibly a stupid question, but I'm wondering if there is GAP code
> which implements an isomorphism
>
> GF(p)[x]/(f(x)) --> GF(p^d),
>
> where f(x) in GF(p)[x] is irreducible.

What you would have to do is to calculate a root a of f in GF(p^d), thsn you can
just evaluate at a:

gap> x:=X(GF(3),"x");
x
gap> f:=x^7+x^3+x+1;
x^7+x^3+x+Z(3)^0
gap> IsIrreducible(f);
true
gap> RootsOfUPol
gap> RootsOfUPol(GF(3^7),f);
[ Z(3^7)^1551, Z(3^7)^281, Z(3^7)^343, Z(3^7)^517, Z(3^7)^843, Z(3^7)^901,
Z(3^7)^1029 ]
gap> a:=last[1]; # for example - you could take any other
Z(3^7)^1551
gap> Value(f,a);
0*Z(3)
gap> pol:=x^2+x+1;
x^2+x+Z(3)^0
gap> Value(pol,a);
Z(3^7)^74

All the best,

Alexander Hulpke

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

```