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## Re: [Axiom-developer] About finite fields

 From: Ralf Hemmecke Subject: Re: [Axiom-developer] About finite fields Date: Mon, 28 Oct 2013 10:19:39 +0100 User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:24.0) Gecko/20100101 Thunderbird/24.0

```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

```