[GAP Forum] subfields of big finite fields

[Alexander Hulpke]

Dear Forum, Dear Inneke,

> 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:
> 
> gap> Subfields(F);
> Error, no method found! For debugging hints type ?Recovery from

Indeed (while we know their orders from theory) GAP currently does not have a method for giving subfields in terms of a user-selected primitive root. Such a method will be added in the next release.

In case it is of help for your calculation, here are (calculated in the development version) the subfields in terms of your primitive root:

gap> Subfields(F);
[ Field([!Z(2)^0]), Field([a^17+a^16+a^11+a^9+a^7+a^6+a^5+a^4+a]), 
  Field([a^15+a^13+a^10+a^9+a^6+a^4+Z(2)^0]), 
  Field([a^17+a^16+a^15+a^13+a^11+a^10+a^9+a^7+a^6+a^5+a^4+a]), 
  Field([a^17+a^13+a^12+a^11+a^10+a^9+a^8+a^7+a^6+a^2+Z(2)^0]), 
  <algebra-with-one of dimension 18 over GF(2)> ]

I hope this is of help,

  Alexander


-- 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