Re: [Axiom-developer] About finite fields

[Ralf Hemmecke]

On 10/28/2013 10:04 AM, Fabio S. wrote:
>> The problem with having Axiom "automatically" perform a
>> irreducibility check is that the check could be time consuming.

That's true.

>> Suppose you have a polynomial of high degree whose factorization
>> takes, say, a couple of minutes.  With an automatic irreducibility
>> test, AXIOM would factor the polynomial every time you create the
>> finite field, which is probably not what you would want.

I haven't checked Axiom, but in FriCAS (I guess the documentation is
still identical to the one in Axiom) it says for
FiniteFieldExtensionByPolynomial:

  ... generated by the extension polynomial defpol which
  MUST be irreducible.

However, if the parameters given to a domain D are identical, then Axiom
should be smart enough to take the already created domain. In other
words if you have

  a: FiniteExtensionFieldByPolynomial(F, defpol) := something
  b: FiniteExtensionFieldByPolynomial(F, defpol) := something_else

then Axiom would create the extension domain only once.

OK, I haven't checked this, but this is the idea.

Of course there can be situations where
FiniteExtensionFieldByPolynomial(F, defpol) would be used in a
completely different context so that Axiom would have no chance to
realize that this domain has already been computed, then it would create
a second copy.

Best thing probably is to do

E := FiniteExtensionFieldByPolynomial(F, defpol)
a: E := ...
b: E := ...

Ralf