On 29 February 2016 at 14:19, Ben Hutz <bn4...@gmail.com> wrote:
> I was exploring some quotient ring operations and came across the following:
>
> {{{
> R.<y>=QQ[]
> K.<w>=NumberField(y^3 + 2*y - 2401)
> k.<v>=K.quo(K.prime_factors(7)[1])
> R.<x,y>=PolynomialRing(k)
> R.monomial_divides(y,x^3*y)
> Error
> }}}
>
> {{{
> R.<y>=QQ[]
> K.<w>=NumberField(y^3 + 2*y - 2401)
> k.<v>=K.residue_field(K.prime_factors(7)[1])
> R.<x,y>=PolynomialRing(k)
> R.monomial_divides(y,x^3*y)
> True
> }}}
>
> The second works, the first does not. I came across this since
> .monomial_divides is used is a groebner basis computation. So played around
I am not sure what you are expecting to get by doing K.quo(P) where P
is an ideal of the ring oif integers of K. The result seems to know
lottle about itself and have no functionality, while
K.residue_field(P) does what you would expect.
> a little more and the following seems weirder
>
> {{{
> A.<x,y,z> = PolynomialRing(Zmod(42))
> A.monomial_divides(y,x^2*y)
> True
> }}}
>
> {{{
> A.<x,y,z> = PolynomialRing(Zmod(2521515232)) #but one less digit works
> A.monomial_divides(y,x^2*y)
> Error
> }}
>
> The first works, but the 2nd does not, even though neither is prime. It
> seems to have to do with what type of polynomial ring they are initialized
> as, but I had a hard time tracking down where that code lived and how it
> decided. Is there someone familiar with polynomial rings in Sage who could
> shed some light on whether this is expected behavior?
I don't know about this one but do not find it surprising that Zmod(n)
should give a different type of object depending on the value of n.
For one thing, n may or may not be prime, and it may or may not b
sensible to take the time to check that.
John
>
> Thanks,
> Ben
>
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