Re: [Axiom-mail] Expressions over finite fields

[root]

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