Re: [Axiom-developer] Issue 351

[Martin Rubey]

Dear Waldek,

Waldek Hebisch <address@hidden> writes:

> I wonder what shall we do with issue 351.

[...]

> One possibility is to prevent creation of fake fields: when creating COMPLEX
> F we may try to check if x^2 - 1 is irreducible over F.  In general this may
> be hard to check, but just looking at characteristic we can reject PF 5
> cheaply.

I just tried to find out when x^2+1 factors in a finite field, but it seems
that the characteristic is not enough to determine this.  

It seems that

* For prime fields with characteristic p, x^2+1 factors if and only if 
  p mod 4 = 1

* More generally, for finite fields with p^m elements and m odd, x^2+1 factors
  if p mod 4 = 1, if m is even, x^2+1 always factors.

Maybe for infinite fields we just try to factor(x^2+1) and check the number of
factors.

However, there is a problem: currently, SPAD does not allow conditional exports
being triggered by functions, but only by "has" and "is" statements, it seems.
At least, I couldn't find an example, nor could I make up one.

:-(

Martin