Re: [Axiom-developer] A quick question about finite fields and

[Waldek Hebisch]

Alasdair McAndrew wrote:
> 
> Hello,
> 
> I'm just looking at axiom again after some years, and I've run into a
> little conundrum.  I have two objects, an element u of the finite field
> FF(7,4), and the defining polynomial f of that field.  Thus:
> 
> (67) -> u
> (67) ->
>             3      2
>    (67)  5%A  + 5%A  + %A
>                                                        Type:
> FiniteField(7,4)
> 68) -> f
> (68) ->
>           4
>    (68)  ?  + ? + 1
>                                 Type: SparseUnivariatePolynomial PrimeField
> 7
> 
> What I want to do is to coerce both objects to polynomials over PF 7, so
> that I can factorize the result.  I want something like
> 
> (69) -> factor(5*b^3+5*b^2+b+b^4+b+1::POLY PF 7)
> (69) ->
>    (69)  (b + 1)(b + 2)(b + 3)(b + 6)
>                                        Type: Factored Polynomial PrimeField
> 7
> 
> But how can I obtain the polynomial in expression (69) without typing it
> in; that is, from u and f directly?

In interpreter that is easy:

(1) -> fF := FiniteField(7,4)                                      

   (1)  FiniteField(7,4)
                                                                   Type: Type
(2) -> pF := PrimeField(7)              

   (2)  PrimeField(7)
                                                                   Type: Type
(3) -> u := (5*b^3+5*b^2+b+b^4+b+1)::SparseUnivariatePolynomial(fF)

         4     3     2
   (3)  ?  + 5?  + 5?  + 2? + 1
                           Type: SparseUnivariatePolynomial(FiniteField(7,4))
(4) -> u::SparseUnivariatePolynomial(pF)                           

         4     3     2
   (4)  ?  + 5?  + 5?  + 2? + 1
                              Type: SparseUnivariatePolynomial(PrimeField(7))

In Spad you need to use:

   map((c : fF) : pF +-> retract(c)@pF, 
u)$UnivariatePolynomialCategoryFunctions2(fF, SUP fF, pF, SUP pF)

In more complex cases you may need PolynomiaCategoryLifting.

-- 
                              Waldek Hebisch
address@hidden