> < ^ From:

< ^ Subject:

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

> < [top]