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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

  ... 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 := ...


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