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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