[GAP Forum] subfields of big finite fields

[Bill Allombert]

On Mon, Nov 26, 2012 at 04:57:38PM +0100, Inneke Van Gelder wrote:
> Dear GAP forum,
> 
> 
> I extend GF(2) by a primitive 19-th root of unity as follows:
> 
> gap> xi:=PrimitiveRoot(GF(2^18))^((2^18-1)/19);
> z
> gap> F:=GF(GF(2),MinimalPolynomial(GF(2),xi));
> <field of size 262144>
> 
> However I am interested in the subfields of this field. But this causes
> an error in GAP:
> 
> Is there a way to get around this error?

Galois theory predict that the set of subfields is exactly 
{GF(2^d) for all d dividing 18} 
and you can even write a generator of each fields in term of a
power of PrimitiveRoot(GF(2^18)) as 
PrimitiveRoot(GF(2^18))^((2^18-1)/(2^d-1)).

Cheers,
Bill.