On Fri, Oct 21, 2011 at 3:49 PM, Chris Smith <[email protected]> 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
In [42]: FF(4)(1).invert()
Out[42]: 1 mod 4
In [43]: FF(4)(2).invert()
NotInvertible: zero divisor
In [44]: FF(4)(3).invert()
Out[44]: 3 mod 4
1st and 3rd answer doesn't make any sense as inverse doesn't exist at all.
I second Aaron's suggestions of making class FiniteRing and put more
constrains on FintiteField. So, the output **would** be something like -
>>> 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?
>
This looks good. Even it can be extended to non linear polynomials using
CRT.
@smichr : Did you pull the function you once mentioned in the pull request #
390 [1] for solving system of congruence relations where CRT fails? That
function can be used to do the above task.
[0] http://en.wikipedia.org/wiki/Field_(mathematics)
[1] https://github.com/sympy/sympy/pull/390#issuecomment-2402863
>
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--
-Regards
Hector
Whenever you think you can or you can't, in either way you are right.
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