Dear Simon,
Thank you very much for your help.
Regards,
Santanu
On 26 February 2017 at 00:03, Simon King <simon.k...@uni-jena.de> wrote:
> Hi Santanu,
>
> I am sorry that your question was unanswered for so long.
>
> On 2017-02-24, Santanu Sarkar <sarkar.santanu....@gmail.com> wrote:
> > How to check $x+4 \in <1+x+x^2+2x^3>$ in the ring $\mathbb{Z}_8[x]$,
> where
> ><1+x+x^2+2x^3> is the ideal generated by 1+x+x^2+2x^3?
> > If yes, how to find $g(x)$ so that $g(x) (1+x+x^2+2x^3)=x+4$?
>
> Here, Singular (or libsingular) can help. Singular provides the method
> "lift". You are working in ZZ/8[x], which is a quotient of ZZ[x]. Hence,
> your relation ideal has the two generators 1+x+x^2+2x^3 and 8.
>
> A slight complication: The default implementation of ZZ[x] does not use
> Singular. Hence, below I implicitly force using singular by defining a
> *multivariate* polynomial ring over ZZ:
> sage: R.<x,y> = ZZ[]
> sage: I = [1+x+x^2+2*x^3, 8]*R
>
> And then you can check containment in a straight forward way:
> sage: 6*x^2 + 6*x + 2 in I
> True
> sage: x+4 in I
> False
>
> So, it should be possible to express 6*x^2+6*x+2 in terms of 8 and
> 1+x+x^2+2*x^3. Indeed, using Singular's lift, we obtain this:
> sage: from sage.libs.singular.function import singular_function
> sage: lift = singular_function('lift')
> sage: L = lift(I, 6*x^2 + 6*x + 2); L
> [24*x^2 - 12*x - 6]
> [ -6*x^5 + 3*x + 1]
> sage: L[0]*I.0+L[1]*I.1
> (6*x^2 + 6*x + 2)
> sage: lift(I, x+4)
> Traceback (most recent call last):
> ...
> RuntimeError: error in Singular function call 'lift':
> 2nd module does not lie in the first
>
> Best regards,
> Simon
>
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