> < ^ Date: Fri, 12 Aug 1994 11:33:00 +0100 (WET)
> < ^ From: Thomas Breuer <Thomas.Breuer@Math.RWTH-Aachen.DE >
< ^ Subject: Re: Galois Field

Dear Mrs. and Mr. Forum,

in his message of today Josep M. Arques writes

I am new to GAP and I am interested in using it to compute in Galois's
Fields with a large number of elements.
I have found that GAP has te limit of 2^16 elements and I would like
to know if there is a way to expand this capability.

The limit of 2^16 results from the internal representation of finite field
elements in GAP, so there is no trick to construct bigger finite fields.

Of course it is possible to choose an irreducible polynomial f of degree n
over the field GF(q), say, and then use the polynomials over GF(q) modulo
f as elements of the field GF( q^n ).

But this is not really a realization of big finite fields in GAP, since
there are up to now no data structures such as ideals and cosets that
would allow to regard an object g + (f) as element of GF(q)[X] / (f)
and thus as a field element. Consequently for this construction one cannot
use GAP functions for fields (such as for the computation of a basis, a
primitive root, or the Galois group) or their elements (such as for the
computation of trace or norm).

In other words, up to now there is no support by data structures and
functions in GAP for handling finite fields of size bigger than 2^16, or
for handling their elements. In the (not too near) future, however, the
limit of 2^16 will be extended.

Kind regards
Thomas Breuer

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