Le samedi 22 octobre 2011 à 11:05 +0530, Hector a écrit :
>
>
> On Fri, Oct 21, 2011 at 3:49 PM, Chris Smith <smi...@gmail.com> wrote:
> I just noticed in Integer, too, the invert method which gives
> the
> multiplicative inverse mode n of a number:
>
> >>> invert(S(3),5)
> 2
> >>> (3*2)%5 == 1
> True
> >>> invert(S(4),6)
> Traceback (most recent call last):
> File "<stdin>", line 1, in <module>
>
> File "sympy\polys\polytools.py", line 4249, in invert
> return f.invert(g)
> File "sympy\core\numbers.py", line 1322, in invert
> raise ZeroDivisionError("zero divisor")
> ZeroDivisionError: zero divisor
>
>
> Before tackling this problem, we need to answer - is ``invert`` (or
> inverse in finite field) in its present form valid??
>
> In [39]: FF?
> Docstring :
> Finite field based on Python's integers.
>
> In [40]: FF(4)
> Out[40]: ℤ₄
>
> But FF(4) fails to be a field [0]. Multiplication is not well defined.
> This thing is not highlighted by sympy
But mathematically, FF(4) exists and is a field. However, FF(4) != ZZ_4.
> I second Aaron's suggestions of making class FiniteRing and put more
> constrains on FintiteField. So, the output **would** be something
> like
FiniteRing isn't a good name, because there can be many finite rings of
a given order. "CyclicRing" or "CongruenceRing" would be unambiguous.
>
> >>> FF(4)
> Error - Z4 is not a filed
>
> >>> FR(4)
> Z4
>
> >>> FR(4)(3).invert()
> 1 ## Additive inverse
>
> >>> FF(5)
> Z5
>
> >>> FF(5)(4).invert_mul()
> 4
>
> >>> FF(5)(4).invert_add()
> 1
>
>
>
> To find an Integer given a description of it's modulus profile
> perhaps
> a method "from_moduli" could be added so
> Integer.from_moduli((2, 3),
> (3, 5)) would give 8. Hector, Mateusz, anyone else: does that
> look
> like a good way to construct it? Is there a better method
> name?
That's the Chinese remainder problem[1]. The solution isn't unique so I
don't think this should be a classmethod of Integer.
[1]: http://en.wikipedia.org/wiki/Chinese_remainder_theorem
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