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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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