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. <s...@unife.it> 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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