#6046: [with patch, needs review] Implement local and global heights for number
field elements
---------------------------+------------------------------------------------
Reporter: cremona | Owner: was
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.0.1
Component: number theory | Keywords: number field height
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Comment(by fwclarke):
Replying to [comment:2 cremona]:
> Your comments and fixes for numerator/denominator ideals are *wrong*!
number field elements have numerators and denominators which are dependent
on the basis used to represent them, and are not what I meant or need --
as I thought my doctest made clear!
I don't see how this can be. The functions I'm calling are not basis-
dependent.
They are essentially the same as yours, but for ideals rather than
elements, and the code is nearly identical.
E.g., for `denominator_ideal(self)` (leaving aside the non-zero check) you
do
{{{
K = self.number_field()
one = K.ideal(1)
return one / (one + K.ideal(self))
}}}
while I've suggested
{{{
return self.number_field().ideal(self).denominator()
}}}
and for a fractional ideal `self` the `denominator` function returns
{{{
try:
return self._denom_ideal
except AttributeError:
pass
self._denom_ideal = (self +
self.number_field().unit_ideal())**(-1)
return self._denom_ideal
}}}
The case for using the ideal `denominator` and `numerator` functions is, I
think,
(i) the general preference for not implementing the same thing twice;
(ii) the fact that the functions in `number_field_ideal.py` cache their
output.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6046#comment:3>
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