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Submitted by : (unknown) at: 2007-11-17T22:32:05-08:00 (9 years ago)
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Axiom believes

Complex PF 5 has Field

\label{eq1} \mbox{\rm true} (1)
Type: Boolean

but this is not true, as Waldek observed:

(2 + %i)::COMPLEX PF 5 *(2 - %i)

Type: Complex(PrimeField?(5))

In fact, we find in gaussian.spad:

     if R has Field then        -- this is a lie; we must know that
       Field                    -- x**2+1 is irreducible in R

Waldek suggested: 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 think that's better than nothing. Maybe Axiom should issue a warning when it cannot determine whether x^2-1 is irreducible?

Is it really hard to check?


Meanwhile I noticed that for a finite field F of characteristic greater than two, x^2+1 seems to be irreducible if and only if the size of the field is congruent three modulo four. A colleague of mine said that good algorithms should exist for most interesting fields, and can probably be found in von zur Ganthen.

However, in SPAD we currently cannot use that knowledge: conditions for exports must be of the form D has C or D1 is D2. It doesn't seem to be possible to call a function. Hopefully this can be changed soon.


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