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#353 case sensitive file names ←	↑	→ #355 Wrong sign in derivation of besselK

#354 Complex R is not necessarily a field when R is a field

Edit detail for #354 Complex R is not necessarily a field when R is a field revision 1 of 1

	1
Editor: Time: 2007/11/17 22:32:05 GMT-8
Note: Remarks

changed:
-
Axiom believes

\begin{axiom}
Complex PF 5 has Field
\end{axiom}

but this is not true, as Waldek observed:

\begin{axiom}
(2 + %i)::COMPLEX PF 5 *(2 - %i)
\end{axiom}

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?

Martin


From kratt6 Sat May 19 08:28:49 -0500 2007
From: kratt6
Date: Sat, 19 May 2007 08:28:49 -0500
Subject: Remarks
Message-ID: <20070519082849-0500@wiki.axiom-developer.org>

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.

Martin

Axiom believes

axiom

Complex PF 5 has Field

$\label{eq1} \mbox{\rm true}$

(1)

Type: Boolean

but this is not true, as Waldek observed:

axiom

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

$\label{eq2}0$

(2)

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 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 is irreducible?

Is it really hard to check?

Martin

Remarks --kratt6, Sat, 19 May 2007 08:28:49 -0500 reply

Meanwhile I noticed that for a finite field F of characteristic greater than two,

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.

Martin