Okay, but is that commonly used terminology? I always thought that the word "generator" when used in the context of finite fields meant a generator of the group of non-zero elements of the field (i.e. a primitive element of the field).
Paul On Fri, 24 Aug 2018 at 00:08, Prof. Dr. Johannes Grabmeier privat <[email protected]> wrote: > > no bug! > > generator() does not necessarily return a primitive element, which > generates the cyclic group (F-{0},*)! It returns an element which generates > the algebra, so the result is correct. > > See comment: > > generator : () -> % > ++ generator() returns a root of the defining polynomial. > ++ This element generates the field as an algebra over the ground field. > > Perhaps you want to use > > primitiveElement()$F16 > > (84) -> w := primitiveElement()$FFP(PF 2, p) > > (84) %E + 1 > Type: > FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1) > (85) -> order w > > (85) 15 > > Type: PositiveInteger > > > > > Am 23.08.18 um 23:22 schrieb Paul Onions: > > Hi All, > > > > It seems that calling generator() on a field created with > > FiniteFieldExtensionByPolynomial may not return a primitive element of > > the field. Specifically, when the field is created using an > > irreducible (but not primitive) polynomial. > > > > Example:- > > > > (1) -> p : UP(x,PrimeField(2)) := x^4 + x^3 + x^2 + x + 1 > > > > 4 3 2 > > (1) x + x + x + x + 1 > > Type: > > UnivariatePolynomial(x,PrimeField(2)) > > (2) -> F16 := FiniteFieldExtensionByPolynomial(PrimeField(2), p) > > > > (2) FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1) > > Type: > > Type > > (3) -> g := generator()$F16 > > > > (3) %A > > Type: > > FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1) > > (4) -> g^5 > > > > (4) 1 > > Type: > > FiniteFieldExtensionByPolynomial(PrimeField(2),?^4+?^3+?^2+?+1) > > > > Looking at the source (ffp.spad) it looks like the implementation of > > generator() assumes that the defining polynomial is primitive, but the > > comments at the head of the file clearly state that the only > > requirement on the defining polynomial is that it be irreducible. > > > > Am I misunderstanding something here? > > Paul > > > > -- > Mit freundlichen Grüßen > > Johannes Grabmeier > > Prof. Dr. Johannes Grabmeier > Köckstraße 1, D-94469 Deggendorf > Tel. +49-(0)-991-2979584, Tel. +49-(0)-151-681-70756 > Tel. +49-(0)-991-3615-141 (d), Fax: +49-(0)-32224-192688 > > -- > You received this message because you are subscribed to the Google Groups > "FriCAS - computer algebra system" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/fricas-devel. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/fricas-devel. For more options, visit https://groups.google.com/d/optout.
