QQ.number_field() does not behave like any other NumberField #7596
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comment:3
sorry modified the wrong ticket, i meant to edit #9414 which is a duplicate of this one |
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Attachment: trac_7596_places_for_QQ.patch.gz |
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comment:5
here is already a ticket for the easy part : a method "places" for QQ |
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comment:6
I have imported a patch trying to have QQ and number fields in the same categories: but this breaks the lcm and gcd in a bad way.. |
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comment:11
As of 9.3.beta3, certainly QQ.places() does work. What I usually do is, when I find a method which QQ does not have but a general number field does have, in the course of implementing something, I just add an aimplementation for QQ. I think this is more efficient than trying to make QQ work like a number field. There will be a large number of users who get to see QQ who never use more general number fields. For such people it does not matter at all to have methods like QQ.places() whose use they may not know. There were many arguments about what QQ.ideal(5) should return, since the pedantic aswer (as for any field, with a nonzero generator) is that it has to be "Principal ideal (1) of ...". Number theorists are quite happy to have K.ideal(...) meaning K.fractional_ideal(...). Is there any reason not to mark this as invalid/won't fix? |
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comment:13
-1 for closing. I think comment:11 discloses a serious problem, though not exactly what is emphasized in the ticket description. Number fields are rings: Thus, the I think that if number theorists are not willing to accept this, then |
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comment:14
We have had this discussion many times over the years. I think I first had it in Jan 2008. There are correct term for these: "integral ideals" and "fractional ideals". Calling them "ideals" is an abuse of language (agreed), just a common one. There is no reason at all not to use "fractional_ideal" as the method name -- the class name is already This ticket has been open for 11 years... |
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comment:15
I think I can live with the practical effect of the implementation of so its documentation directly contradicts its behaviour. This is due to the fact that the method is inherited from Reassigning this ticket to a milestone, because having documentation directly contradict behaviour is obviously something that not just generic "low priority" or "not a bug". |
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comment:17
Stalled in |
This was referenced Aug 2, 2021
vbraun
pushed a commit
to vbraun/sage
that referenced
this issue
Jan 29, 2024
… `.discriminant()` and `.is_isomorphic` of quaternion algebras
1. Removed unnecessary product and factorization for
`.ramified_primes()`
2. Adapted `is_isomorphic` to reduce unnecessary calculations
3. Fixed a bug in `.discriminant()` and `.ramified_primes()` where
rational invariants caused errors
4. Removed `.hilbert_conductor` from `sage.arith.misc` import list and
added `.hilbert_symbol`
5. Reduced restriction of `.ramified_primes()` to rational quaternion
algebras in docstring in preparation for planned `.ramified_places()`
function over number fields (currently being worked on)
In more detail:
1. The original workflow for `.ramified_primes()` went as follows:
- Call `.discriminant()`, which calls `.hilbert_conductor`
- Inside `.hilbert_conductor`, the ramified primes are computed and
their product (the discriminant of the quaternion algebra) is returned
- Finally, factor the discriminant back into its prime factors
Hence we have a redundant product and, more crucially, a redundant
prime factorization. This fix modifies `.ramified_primes()` to instead
directly build the list computed in `.hilbert_conductor()` (up to a bug
fix described in 3.) and return it; the list might not always be sorted
by magnitude of primes, so an optional argument `sorted` (set to `False`
by default) allows to enforce this (small to large) sorting.
Furthermore, `.discriminant()` has been adapted to directly take the
product of the list returned by `.ramified_primes()` (only in the
rational case, for now - see 5.)
2. Since the `.discriminant()`-function needs to compute all (finite)
ramified primes (this was also true before this PR, it was just hidden
inside `.hilbert_conductor()` instead), the function `.is_isomorphic()`
now compares the unsorted lists of finite ramified primes to decide
whether two rational quaternion algebras are isomorphic.
3. The function `sage.arith.misc.hilbert_conductor` requires its
arguments to be integers (to create certain lists of prime divisors);
since it was originally used to determine the discriminant (and, as
explained in 1., the ramified primes), it raises an error when the
invariants are proper rational numbers. To get around the analogous
error for the method `.hilbert_symbol`, we instead look at the
numerators and denominators of both invariants separately, using the
fact that we can (purely on a mathematical level) rescale both
invariants by the squares of their respective denominators without
leaving the isomorphism class of the algebra.
4. The only call to `sage.arith.misc.hilbert_conductor` in
quaternion_algebra.py was given in the old computation of the
discriminant (the other `.hilbert_conductor` in the code, also in
`.discriminant()`, refers to the one in `sage.rings.number_field`), so
it was removed from the import list. The new approach to
`.ramified_primes()` requires `sage.arith.misc.hilbert_symbol`, which
was added to the import list.
5. As of now the `.ramified_primes()`-method is only supported for
rational quaternion algebras. I'm currently working on a version over
number fields, but once it works correctly this will be implemented as a
new function `.ramified_places` (Update: see sagemath#37173) ~~to distinguish
between different formats (prime numbers vs ideals) over $\mathbb{Q}$~~
(Update: this wasn't really feasible, see the issues discussed in sagemath#7596;
thanks to @yyyyx4 for pointing me towards this discussion) ~~and,
furthermore,~~ to not cause confusion using the term "primes" for the
Archimedean real places where a quaternion algebra might ramify. Hence
the implementation restriction in the docstring of `.ramified_primes()`
was removed, but the method still throws a ValueError if not called with
a quaternion algebra defined over the rational numbers.
#sd123
URL: sagemath#37164
Reported by: Sebastian Spindler
Reviewer(s): grhkm21, Sebastian Spindler
5 tasks
vbraun
pushed a commit
to vbraun/sage
that referenced
this issue
Jan 30, 2024
… `.discriminant()` and `.is_isomorphic` of quaternion algebras
1. Removed unnecessary product and factorization for
`.ramified_primes()`
2. Adapted `is_isomorphic` to reduce unnecessary calculations
3. Fixed a bug in `.discriminant()` and `.ramified_primes()` where
rational invariants caused errors
4. Removed `.hilbert_conductor` from `sage.arith.misc` import list and
added `.hilbert_symbol`
5. Reduced restriction of `.ramified_primes()` to rational quaternion
algebras in docstring in preparation for planned `.ramified_places()`
function over number fields (currently being worked on)
In more detail:
1. The original workflow for `.ramified_primes()` went as follows:
- Call `.discriminant()`, which calls `.hilbert_conductor`
- Inside `.hilbert_conductor`, the ramified primes are computed and
their product (the discriminant of the quaternion algebra) is returned
- Finally, factor the discriminant back into its prime factors
Hence we have a redundant product and, more crucially, a redundant
prime factorization. This fix modifies `.ramified_primes()` to instead
directly build the list computed in `.hilbert_conductor()` (up to a bug
fix described in 3.) and return it; the list might not always be sorted
by magnitude of primes, so an optional argument `sorted` (set to `False`
by default) allows to enforce this (small to large) sorting.
Furthermore, `.discriminant()` has been adapted to directly take the
product of the list returned by `.ramified_primes()` (only in the
rational case, for now - see 5.)
2. Since the `.discriminant()`-function needs to compute all (finite)
ramified primes (this was also true before this PR, it was just hidden
inside `.hilbert_conductor()` instead), the function `.is_isomorphic()`
now compares the unsorted lists of finite ramified primes to decide
whether two rational quaternion algebras are isomorphic.
3. The function `sage.arith.misc.hilbert_conductor` requires its
arguments to be integers (to create certain lists of prime divisors);
since it was originally used to determine the discriminant (and, as
explained in 1., the ramified primes), it raises an error when the
invariants are proper rational numbers. To get around the analogous
error for the method `.hilbert_symbol`, we instead look at the
numerators and denominators of both invariants separately, using the
fact that we can (purely on a mathematical level) rescale both
invariants by the squares of their respective denominators without
leaving the isomorphism class of the algebra.
4. The only call to `sage.arith.misc.hilbert_conductor` in
quaternion_algebra.py was given in the old computation of the
discriminant (the other `.hilbert_conductor` in the code, also in
`.discriminant()`, refers to the one in `sage.rings.number_field`), so
it was removed from the import list. The new approach to
`.ramified_primes()` requires `sage.arith.misc.hilbert_symbol`, which
was added to the import list.
5. As of now the `.ramified_primes()`-method is only supported for
rational quaternion algebras. I'm currently working on a version over
number fields, but once it works correctly this will be implemented as a
new function `.ramified_places` (Update: see sagemath#37173) ~~to distinguish
between different formats (prime numbers vs ideals) over $\mathbb{Q}$~~
(Update: this wasn't really feasible, see the issues discussed in sagemath#7596;
thanks to @yyyyx4 for pointing me towards this discussion) ~~and,
furthermore,~~ to not cause confusion using the term "primes" for the
Archimedean real places where a quaternion algebra might ramify. Hence
the implementation restriction in the docstring of `.ramified_primes()`
was removed, but the method still throws a ValueError if not called with
a quaternion algebra defined over the rational numbers.
#sd123
URL: sagemath#37164
Reported by: Sebastian Spindler
Reviewer(s): grhkm21, Sebastian Spindler
vbraun
pushed a commit
to vbraun/sage
that referenced
this issue
Feb 1, 2024
… `.discriminant()` and `.is_isomorphic` of quaternion algebras
1. Removed unnecessary product and factorization for
`.ramified_primes()`
2. Adapted `is_isomorphic` to reduce unnecessary calculations
3. Fixed a bug in `.discriminant()` and `.ramified_primes()` where
rational invariants caused errors
4. Removed `.hilbert_conductor` from `sage.arith.misc` import list and
added `.hilbert_symbol`
5. Reduced restriction of `.ramified_primes()` to rational quaternion
algebras in docstring in preparation for planned `.ramified_places()`
function over number fields (currently being worked on)
In more detail:
1. The original workflow for `.ramified_primes()` went as follows:
- Call `.discriminant()`, which calls `.hilbert_conductor`
- Inside `.hilbert_conductor`, the ramified primes are computed and
their product (the discriminant of the quaternion algebra) is returned
- Finally, factor the discriminant back into its prime factors
Hence we have a redundant product and, more crucially, a redundant
prime factorization. This fix modifies `.ramified_primes()` to instead
directly build the list computed in `.hilbert_conductor()` (up to a bug
fix described in 3.) and return it; the list might not always be sorted
by magnitude of primes, so an optional argument `sorted` (set to `False`
by default) allows to enforce this (small to large) sorting.
Furthermore, `.discriminant()` has been adapted to directly take the
product of the list returned by `.ramified_primes()` (only in the
rational case, for now - see 5.)
2. Since the `.discriminant()`-function needs to compute all (finite)
ramified primes (this was also true before this PR, it was just hidden
inside `.hilbert_conductor()` instead), the function `.is_isomorphic()`
now compares the unsorted lists of finite ramified primes to decide
whether two rational quaternion algebras are isomorphic.
3. The function `sage.arith.misc.hilbert_conductor` requires its
arguments to be integers (to create certain lists of prime divisors);
since it was originally used to determine the discriminant (and, as
explained in 1., the ramified primes), it raises an error when the
invariants are proper rational numbers. To get around the analogous
error for the method `.hilbert_symbol`, we instead look at the
numerators and denominators of both invariants separately, using the
fact that we can (purely on a mathematical level) rescale both
invariants by the squares of their respective denominators without
leaving the isomorphism class of the algebra.
4. The only call to `sage.arith.misc.hilbert_conductor` in
quaternion_algebra.py was given in the old computation of the
discriminant (the other `.hilbert_conductor` in the code, also in
`.discriminant()`, refers to the one in `sage.rings.number_field`), so
it was removed from the import list. The new approach to
`.ramified_primes()` requires `sage.arith.misc.hilbert_symbol`, which
was added to the import list.
5. As of now the `.ramified_primes()`-method is only supported for
rational quaternion algebras. I'm currently working on a version over
number fields, but once it works correctly this will be implemented as a
new function `.ramified_places` (Update: see sagemath#37173) ~~to distinguish
between different formats (prime numbers vs ideals) over $\mathbb{Q}$~~
(Update: this wasn't really feasible, see the issues discussed in sagemath#7596;
thanks to @yyyyx4 for pointing me towards this discussion) ~~and,
furthermore,~~ to not cause confusion using the term "primes" for the
Archimedean real places where a quaternion algebra might ramify. Hence
the implementation restriction in the docstring of `.ramified_primes()`
was removed, but the method still throws a ValueError if not called with
a quaternion algebra defined over the rational numbers.
#sd123
URL: sagemath#37164
Reported by: Sebastian Spindler
Reviewer(s): grhkm21, Sebastian Spindler
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Several examples:
This creates ideal that does not have the functions I.denominator, I.numerator, I.prime_ideals() ... which a fractional ideal in a number field should have
Similarly, QQ.places() is not implemented; it should return the one infinite place for Q. Although there seems to be QQ.embeddings().
CC: @slel
Component: number fields
Issue created by migration from https://trac.sagemath.org/ticket/7596
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