Why do Finite Fields have DifferentialRing?

[Prof. Dr. Johannes Grabmeier privat]

Any reasons that FiniteFieldCategory requires all finite fields to be a
DifferentialRing with trivial implementation pf differentiate(x)  = 0? I
woud like to eliminate this.

--
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

[Ralf Hemmecke]

On 1/5/20 8:48 PM, Prof. Dr. Johannes Grabmeier privat wrote:
> Any reasons that FiniteFieldCategory requires all finite fields to be a
> DifferentialRing with trivial implementation pf differentiate(x)  = 0? I
> woud like to eliminate this.

Waldek has already answered that question.

https://groups.google.com/forum/#!topic/fricas-devel/cDim4Gt9PJk

Can you give reasons why you think it would be preferable to remove the
export and its default category implementation?

Ralf

[Prof. Dr. Johannes Grabmeier FW]

I forgot about this (thank you for pointing it out again) and started interrupted work again. But the answer was not what I expected. I totally agree with the principles,

"This is mainy issue of consistency: we build structures by recursion,  so trivial cases are necessary as base case.  Also, we like to  use generic algorithms and they may need such trivial operations.  So definitely they should be present. "

but my question is, which application/generic algorithm right now uses that finite fields are trival  Differential rings? When want to recursively make use of differentiate for UP('x, R), R a commutative ring, then the differentiate should be already in every commutative ring and fields, but it shows up for the first time in FiniteFieldCategory

And: With this principal answer one could add everything to everything, so why not add a trivial total order x < y being false all the time?

And  here is my answer to your new question: Because I would like to remove superfluous functions to make FriCAS more applicable to novices, when one does )who FF(2,5), then one sees so many functions as "differentiate" and then it is hard to spot the crucial ones.

Johannes

Am 06.01.20 um 11:26 schrieb Ralf Hemmecke:

-- 
Mit freundlichen Grüßen

Johannes Grabmeier

Oberbürgermeisterkandidat der FREIEN WÄHLER
Fraktionsvorsitzender 
FREIE WÄHLER, Stadtrat Deggendorf

Prof. Dr. Johannes Grabmeier
Köckstraße 1, D-94469 Deggendorf
Tel. +49-(0)-991-2979584, Tel. +49-(0)-151-681-70756
Fax: +49-(0)-991-2979592

[Waldek Hebisch]

On Mon, Jan 06, 2020 at 03:55:52PM +0100, Prof. Dr. Johannes Grabmeier FW wrote:
>
> I forgot about this (thank you for pointing it out again) and started
> interrupted work again. But the answer was not what I expected. I
> totally agree with the principles,
>
>
> "This is mainy issue of consistency: we build structures by

> recursion,?? so trivial cases are necessary as base case. ??Also, we
> like to?? use generic algorithms and they may need such trivial
> operations.?? So definitely they should be present. "

>
>
>
> but my question is, which application/generic algorithm right now uses

> that finite fields are trival??

Differential polynomials and differential operators. Constant
coefficient operators make sense and need trivial derivative on
base ring. If the drivative was reomoved, one could work around
problems caused by this. But it is simpler to have trivial
derivative...


--
Waldek Hebisch

[Ralf Hemmecke]

> but my question is, which application/generic algorithm right now uses
> that finite fields are trival  Differential rings?

But the question raised by Johannes is also why we do not immediately
equip every commutative ring with a trivial differential structure?
That can be overridden by the implementation of a concret domain if
necessary.

UnivariatePolynomialCategory declares itself do be a differential ring
and implicitly assumes that the coefficients are constants. That maybe
efficient, but is not generic.

> And: With this principal answer one could add everything to everything,
> so why not add a trivial total order x < y being false all the time?

That's not consistent. If x < y is false, then y < x or x=y should be
true. https://en.wikipedia.org/wiki/Total_order

I agree, however, with Johannes that seeing ALL exported functions of a
domain is too much information for a beginner. I have no solution to
this problem except for writing good tutorials of how to use a domain.

Ralf