Dear all,
How should be defined division in non-integral domains like Zmod(6)? The
following looks a bit incoherent to me
sage: Z6 = Zmod(6)
sage: a = Z6(4)
sage: b = Z6(2)
sage: b.divides(a)
True
sage: a // b
Traceback (most recent call last):
...
ZeroDivisionError: Inverse does not exist.
The reason for the ZeroDivisionError is because the algorithm tries
to invert b and perform a multiplication.
Reading [wiki] the division is well defined when it is unique. So
raising an error seems appropriate. But I would like to ask what
is the most appropriate
1. do not change anything
2. raise a more meaningful error (eg an ArithmeticError or
ValueError with better message)
3. return an actual quotient
I would vote for 2 and add a method to obtain one quotient or all
possible quotients.
Best
Vincent
[wiki]
https://en.wikipedia.org/wiki/Division_(mathematics)#Abstract_algebra
PS: Maple somehow agrees with [wiki] as it does not define division in
Zmod(6)
> with(Domains);
> show(Zmod(6), operations);
` Signatures for constructor Zmod`
` note: operations prefixed by -- are not available`
` * : (Zmod,Zmod*) -> Zmod`
` * : (Integers,Zmod) -> Zmod`
` + : (Zmod,Zmod*) -> Zmod`
` - : Zmod -> Zmod`
` - : (Zmod,Zmod) -> Zmod`
` 0 : Zmod`
` 1 : Zmod`
` <> : (Zmod,Zmod) -> Boolean`
` = : (Zmod,Zmod) -> Boolean`
` -- Characteristic : Integers`
` Coerce : Integers -> Zmod`
` Index : Integers -> Zmod`
` Input : Expression -> Union(Zmod,FAIL)`
` Inv : Zmod -> Union(Zmod,FAIL)`
` Lookup : Zmod -> Integers`
` Output : Zmod -> Expression`
` Random : () -> Zmod`
` Size : Integers`
` Type : Expression -> Boolean`
` Universe : () -> Zmod*`
` ^ : (Zmod,Integers) -> Zmod`
PPS: people with access to magma are welcome to report!
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