section.htm

Creating an extension of a Prime Field

In this section we define the functions used to create a finite field with p^d elements. We will use the Maple function GF which creates an array of functions and constants relevant to the field.

I will illustrate them below. GF(p,d) chooses an irreducile polynomial, but I had trouble making

the arithmetic work. So I use the version GF(p,d,f) where I choose f to be irreducible and

primitive.

So we need to find a polynomial, f(T), which is irreducible and of degree d over F_p.

We will also insist that the polynomial be primitive. That is, in the multiplicative group of F_p[T]/f(T), the order of T is p^d - 1.

You'll notice I check that degree(f)=d. Randpoly(d,t) should produce a polynomial of degree d, but for some reason when d>40 it sometimes returns a polynomial of lower degree. So I inserted the check to be sure.

> GetIrrPoly := proc(p,d)
local bool, f;

bool:=false;
while bool = false do
f:= Randpoly(d,T) mod p;
if Irreduc(f) mod p and Primitive(f) mod p
and (degree(f)= d) then bool :=true fi;
od;
RETURN(f);
end proc;

The field GF(2^50)

> p:=2; d:= 50;p := 2;

>

> f := GetIrrPoly(p,d);

Next we use the Maple function GF to create the field. GF returns an array of constants and functions that do most of what we want with the finite field.

> FF:= GF(p,d,f);

>

Now we define our primitive element of the finite field. It is the image of T in the quotient space F_p[T]/f(T).
Maple won't let you use T itself, but it will always express things in terms of T not al. I don't get it.

> al := FF[ConvertIn](T);