> < ^ Date: Mon, 29 Aug 1994 12:30:00 +0200
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
< ^ Subject: Re: Finite Algebraic Field Extensions

Dear GAP-Forum,

In his mail, Lewis McCarthy wrote

I've just noticed that GAP currently only handles
finite fields, the field of rationals, and cyclotomic
extensions of the rationals. Are there any plans to
support arbitrary finite algebraic extensions of the
rationals ? This is probably crucial to the project
for which I'm considering GAP.

I would like to make a few remarks on the status of algebraic
field extensions in GAP:

At the moment (this means GAP 3.4) GAP supports a basic implemen-
tation of simple algebraic extensions for the rationals as well
as finite fields. This is done by representing the extension as a
factor of the polynomial ring modulo the ideal generated by an
irreducible polynomial. These fields can be constructed by the
command 'AlgebraicExtension'. Basic arithmetic as well as polyno-
mial arithmetic and factorization is available for these fields.

Unfortunately, the corresponding documentation has been forgotten
in the distribution, it will be added in the first patch, which
is due in the next few days (the patch will be announced in the
GAP-Forum). However the file forum94b.txt (which is in the 'etc'
directory of the GAP distribution) contains some example calcula-
tions in a letter I wrote to the forum (dated 14. Jun 1994).
Representation of algebraic elements is required to be as a poly-
nomial in the primitive element, a change of the basis (which
might be desirable for some computations in algebraic number
theory) is not supported.

At present time these routines do not allow multiple algebraic
extensions. If needed field towers could be implemented easily.

However, due to the method of representation, no assumptions can
be made about embeddings of the algebraic extensions into other
fields (i.e. the complex numbers). This means as well, that no
assumption about which root of the minimal polynomial is taken as
primitive element can be made (i.e. 2^(1/2) or - 2^(1/2) are both
possible).

With some basic algebraic number theory, it is possible to avoid
some of those drawbacks by 'hand calculations', but necessary
computations might become quite tedious.

At present time, we have no plans to expand those features, as
the current aviabilities are sufficient for our uses in group
theory and its related areas and significant problems will arise
when trying to overcome the mentioned drawbacks.

Alexander Hulpke

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