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Re: [Axiom-mail] Expressions over finite fields
From: |
root |
Subject: |
Re: [Axiom-mail] Expressions over finite fields |
Date: |
Fri, 13 Feb 2004 11:20:26 -0500 |
Marcus,
I looked back thru my email logs and apparently I never replied.
Sorry about that.
> I am looking for a way to handle algebraic expressions over finite
> fields that will allow me to take square roots of variables. For
> example, I want something like
>
> (sqrt(a)*w + w^2)::UTS(EXPR INT, w, 0)
>
> but in positive characteristic. There is apparently no such thing as
> EXPR PrimeField 2.
The type UTS(EXPR INT, w, 0) seems to be constructed fine here.
and (sqrt(a)*w + w^2)::UTS(EXPR INT,w,0) also is a correct
construction.
EXPR has the signature:
Expression(R:OrderedSet)
but if we ask about PrimeField(2):
PrimeField(2) has OrderedSet ==> false
so Axiom will not allow you to build the type because
PrimeField(2) lacks the OrderedSet property.
> I am currently using FRAC POLY PrimeField 2, which allows me to form
> rational functions of variables, but not to take square roots. Is there
> a more appropriate domain available?
You can build this type because the signature of POLY is:
PolynomialRing(R:Ring, E:OrderedAbelianMonoid)
and
PrimeField(2) has Ring ==> true
So, on a type basis, your question becomes "Can I find a domain which
will accept the properties of PrimeField(2) and which supports rational
functions of variables".
You can find a lot of information about PrimeField(2) by typing:
)show PrimeField(2)
Unfortunately it does not show that it is a Ring even though Axiom
can decide the question (as above).
I can't find a domain directly for what you want but there are many
polynomial-style domains (see the src/algebra subdirectory and grep
for poly) and one of them might fit your needs. You need to look
at the signatures and then ask 'PrimeField(2) has ' questions to
see if the construction will be valid.
>Is it possible to work with transcendental field extensions?
Yes. I can't give you an example off the top of my head however.
This is Manuel Bronstein's area of expertise.
Tim