Re: [Axiom-developer] About finite fields

[Clifton Williamson]

Fabio,

The problem with having Axiom "automatically" perform a irreducibility check is that the check could be time consuming.  Suppose you have a polynomial of high degree whose factorization takes, say, a couple of minutes.  With an automatic irreducibility test, AXIOM would factor the polynomial every time you create the finite field, which is probably not what you would want.

Clifton

On Thursday, October 24, 2013 7:00 AM, Fabio S. <address@hidden> wrote:

I realized that when constructing finite fields providing a user choosen
polynomial, Axiom doesn't check whether the used polynomial is
irreducible or not.

For example:

(10) -> factor(x^8+x^4+x^2+x+1::PF 2)
(10) ->
            4    3      4    3    2
    (10)  (x  + x  + 1)(x  + x  + x  + x + 1)
                                Type: Factored(Polynomial(PrimeField(2)))
                                                              Time: 0 sec
(11) -> K:=FFP(PF 2,x^8+x^4+x^2+x+1)
(11) ->
    (11)  FiniteFieldExtensionByPolynomial(PrimeField(2),?^8+?^4+?^2+?+1)
                                                          Type: Domain
                                                Time: 0.01 (IN) = 0.01 sec
(12) -> size()$K
(12) ->
    (12)  256
                                                Type: NonNegativeInteger
                                                              Time: 0 sec
(13) -> a:=index(2)$K
(13) ->
    (13)  %A
    Type: FiniteFieldExtensionByPolynomial(PrimeField(2),?^8+?^4+?^2+?+1)
                                              Time: 0.01 (OT) = 0.01 sec
(14) -> minimalPolynomial a
(14) ->
          8    4    2
    (14)  ?  + ?  + ?  + ? + 1
                          Type: SparseUnivariatePolynomial(PrimeField(2))
                                                              Time: 0 sec
(15) -> (a^4+a^3+1)*(a^4+a^3+a^2+a+1)
(15) ->
    (15)  0
    Type: FiniteFieldExtensionByPolynomial(PrimeField(2),?^8+?^4+?^2+?+1)
                                                              Time: 0 sec


I am surprised that FiniteFieldByPolynomial allows a contruction which,
clearly, has no sense and I am wondering if this has to be considered a
bug to be fixed.

Fabio


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