On 15 August 2017 at 18:42, Nils Bruin <nbr...@sfu.ca> wrote:
> On Tuesday, August 15, 2017 at 7:21:03 AM UTC-7, chandra chowdhury wrote:
>>
>> Is it possible to factor polynomials completely over modular ring?
>>
>> Like
>> x = var('x')
>> factor(x^5-x, IntegerModRing(25)['x'])
>> gives
>>
>> (x-1)(x+1)(x^2+1)*x
>
> The second argument is simply ignored here, by the looks of it
>
> sage: factor(x^5-x,"moo")
> (x^2 + 1)*(x + 1)*(x - 1)*x
>
> You could file that as a (mild) bug.
>
> Factorization of Z/25 directly isn't implemented:
>
> sage: R=IntegerModRing(25)
> sage: Rx.<x>=R[]
> sage: factor(x^5-x)
> NotImplementedError: factorization of polynomials over rings with composite
> characteristic is not implemented
>
> Root finding is apparently implemented:
>
> sage: (x^5-x).roots(multiplicities=False)
> [0, 1, 7, 18, 24]
>
> Alternatively, (beccause you're working mod a prime power) you could look at
> p-adic rings:
>
> sage: R=Zp(5,2,"fixed-mod",print_mode="terse")
> sage: Rx.<x>=R[]
> sage: (x^5-x).factor()
> ((1 + O(5^2))*x + (1 + O(5^2))) * ((1 + O(5^2))*x + (7 + O(5^2))) * ((1 +
> O(5^2))*x + (18 + O(5^2))) * ((1 + O(5^2))*x + (24 + O(5^2))) * ((1 +
> O(5^2))*x + (0 + O(5^2)))
>
> The "+O(5^2)" is a p-adic thing that would be nice to suppress here.A relevant point, perhaps, is that polynomials over Z/25Z do not form a unique factorization domain (or even a domain) so even the definition of what factors you might want to see is unclear. Following Nils, for a prime power modulus, a sensible definition is to take an approximation of the p-adic factorization. The for a general modulus the Chinese remainder theorem is probably relevant. To illustrate my non-uniqueness point, modulo 8 we have x^2-1 = (x+1)(x-1) = (x-3)(x+3). John Cremona > > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-support+unsubscr...@googlegroups.com. > To post to this group, send email to sage-support@googlegroups.com. > Visit this group at https://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
