[GAP Forum] Finite field defined with an irreducible polynomial
Anvita
anvita21 at gmail.com
Sat Aug 23 03:25:52 BST 2008
Dear Forum,
When a finite field is defined using an irreducible polynomial, there seems
to be a problem
finding the coefficients of the field's elements in the natural power bases.
For example:
----------------------------------------
gap> x:=Indeterminate(GF(5),"x");
x
gap> pol:=x^7+x^4+x^2-x+Z(5);
x^7+x^4+x^2-x+Z(5)
gap> F:=GF(GF(5),pol);
<field of size 78125>
gap> a:=RootOfDefiningPolynomial(F);
(a)
gap> t:=a^32;
(Z(5)^2+Z(5)*a^2+Z(5)*a^3+Z(5)^2*a^5+a^6)
-----------------------------------------
Even though "t" is displayed as a polynomial in "a", I do not know
how to get hold of the corresponding coefficients, because the field "F"
as a vector space does not have a basis:
--------------------------------------------------------------------------------
gap>IsVectorSpace(F);
true
gap> b:=List([0..6],i->a^i);
[ !Z(5)^0, (a), (a^2), (a^3), (a^4), (a^5), (a^6) ]
gap> Basis(F,b);
Error, no method found! For debugging hints type ?Recovery from
NoMethodFound
Error, no 3rd choice method found for `PrimitiveRoot' on 1 arguments called
from
PrimitiveRoot( F ) called from
Basis( V ) called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
--------------------------------------------------------------------------------
Is there any way around this problem?
Thank you,
Anvita
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