Re: Urgent aid needed
- To: mathgroup at smc.vnet.net
- Subject: [mg16428] Re: [mg16398] Urgent aid needed
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Sat, 13 Mar 1999 02:21:42 -0500
- Sender: owner-wri-mathgroup at wolfram.com
On Thu, Mar 11, 1999, <c.a.e at wanadoo.fr> wrote:
>Hi,
>Here's the subject:
>Let e,p,q be elements of commutative unitary ring such as e^3==-1
>Prove that D==p*e^2-q divides P==e^2*p^3-p^3*q-e^2*q^3+e^2*p*q^3.
>Please write to me at c.a.e at wanadoo.fr
>Thank you in advance
It seem clear that you are a beginner in Mathematica. So first a few
remarks on the basics: == means equal in Mathematica but = means Set.
Also you can't use the name D, it is already reserved for the
differentiation operator. So I am going to change your names to poly and div.
O.K. first we need to enter the definitions:
In[1]:=
poly=e^2*p^3-p^3*q-e^2*q^3+e^2*p*q^3;
div=p*e^2-q;
Next we find a Groebner basis for the ideal generated by div and e^3+1:
In[2]:=
g=GroebnerBasis[{div,e^3+1}]
Out[2]=
3 3 2 2 2 3
{p - q , p + e q, e p + q , e p - q, 1 + e }
The last step is to reduce poly with respect to the Groebner basis:
In[3]:=
PolynomialReduce[poly,g]
Out[3]=
2 2 3 4
{{e - q, e q , 0, 0, -q }, 0}
As you can see the second element, the remainder, is 0, which proves what
you wanted.
Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp/
http://eri2.tuins.ac.jp/