is_unit() gives incorrect result for quotient of number ring

[gh-mathehertogh]

Consider the following example:

sage: version()
'SageMath version 9.3.beta6, Release Date: 2021-01-17'
sage: K.<a> = NumberField(x^3-2)
sage: O = K.maximal_order()
sage: I = O.ideal(4)
sage: Omod4 = O.quotient(I, 'b')
sage: a + I == 1 # Are a and 4 coprime? (no)
False
sage: a_bar = Omod4(a)
sage: a_bar.is_unit() # So this should return False...
True
sage: a_bar.parent()
Quotient of Maximal Order in Number Field in a with defining polynomial x^3 - 2 by the ideal (4)

Clearly the method is_unit() gives the wrong answer in this example.

I checked the implementation:

sage: a_bar.is_unit??
    ...
    def is_unit(self):
        """
        Return True if self is a unit in the quotient ring.
        ...
        """
        if self.__rep.is_unit():
            return True
        from sage.categories.fields import Fields
        if self.parent() in Fields:
            return not self.is_zero()
        try:
            self.__invert__()
            return True
        except ArithmeticError:
            return False
        raise NotImplementedError

When I trace through this, I do not understand what happens. On the first line self.__rep.is_unit() returns True, while my debugger says self has no attribute __rep. When I step into this call, I end up in K.is_field(), which returns True.

I tried checking if for some reason we where executing the line if self.parent() in Fields, but this does not seem to be the case: self.parent() gives Quotient of Maximal Order in Number Field in a with defining polynomial x^3 - 2 by the ideal (4)

[gh-DaveWitteMorris]

self.__rep is another name for self.lift(), which is equal to a.

The lift is erroneously being considered as an element of the field K, instead of as an element of the order O.

sage: a_bar.lift().parent()
Number Field in a with defining polynomial x^3 - 2
sage: Omod4(2).lift().parent()
Number Field in a with defining polynomial x^3 - 2
sage: Omod4(2).is_unit()
True

Possibly related tickets: #12242 and #28552.

[mkoeppe]

Moving to 9.4, as 9.3 has been released.