sagemathgh-37164: Fixes and simplifications for `.ramified_primes()`, `.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()` to distinguish between different
formats (prime numbers vs ideals) over Q 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

sagemathgh-37164: Fixes and simplifications for `.ramified_primes()`, `.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

sagemathgh-37164: Fixes and simplifications for `.ramified_primes()`, `.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

sagemathgh-37173: Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields
    
1. Implemented method `.ramified_places` for quaternion algebras over
number fields. Integrated `.ramified_primes()` into it in the process.
2. Modified `.is_division_algebra()`, `.is_matrix_ring()` and
`.is_isomorphic` to use `.ramified_places` instead of `.discriminant()`,
thus extending them to base number fields.
3. Rerouted `.discriminant()` through `.ramified_places` since the
original call to `.hilbert_conductor` also computed all finite ramified
places.
4. Added `.is_totally_definite()` and moved `is_definite()`.

Some more detail:
1. The new method `.ramified_places` returns all places at which the
quaternion algebra `self` ramifies; this includes the infinite places by
default, but can be reduced to only the finite places with the optional
parameter `inf`. The old version of `.ramified_primes()` from sagemath#37164 has
been integrated into `.ramified_places`, thus setting the former up for
possible future deprecation; currently it calls
`self.ramified_places(inf=False)` for backwards compatibility.

2. `.is_division_algebra()` and `.is_matrix_ring()` now instead check
whether the list of ramified places (finite and infinite) is trivial.
`.is_isomorphic` now compares the set of finite ramified places and,
unless working over $\mathbb{Q}$, the list of infinite ramified places
of both algebras. The latter can be compared as lists since the real
embeddings of the number field are sorted independently of each
algebras' invariants, but the former (probably) need to be compared as
sets since the order of the list depends on the primes above the
respective invariants. The docstring of `.is_isomorphic` (as well as
some of the other docstrings) now includes an example of a non-split
quaternion algebra with trivial discriminant, namely the algebra with
invariants $(-1,-1)$ over the quadratic field $\mathbb{Q}(\sqrt{5})$.

Possible future work:
- Extend functionality to all global fields (of characteristic not equal
to $2$) [UPDATE: Will be done once both this PR and sagemath#37554 have been
merged]
    
URL: sagemath#37173
Reported by: Sebastian A. Spindler
Reviewer(s): AurelPage, Frédéric Chapoton, grhkm21, Matthias Köppe, Sebastian A. Spindler

sagemathgh-37173: Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields
    
1. Implemented method `.ramified_places` for quaternion algebras over
number fields. Integrated `.ramified_primes()` into it in the process.
2. Modified `.is_division_algebra()`, `.is_matrix_ring()` and
`.is_isomorphic` to use `.ramified_places` instead of `.discriminant()`,
thus extending them to base number fields.
3. Rerouted `.discriminant()` through `.ramified_places` since the
original call to `.hilbert_conductor` also computed all finite ramified
places.
4. Added `.is_totally_definite()` and moved `is_definite()`.

Some more detail:
1. The new method `.ramified_places` returns all places at which the
quaternion algebra `self` ramifies; this includes the infinite places by
default, but can be reduced to only the finite places with the optional
parameter `inf`. The old version of `.ramified_primes()` from sagemath#37164 has
been integrated into `.ramified_places`, thus setting the former up for
possible future deprecation; currently it calls
`self.ramified_places(inf=False)` for backwards compatibility.

2. `.is_division_algebra()` and `.is_matrix_ring()` now instead check
whether the list of ramified places (finite and infinite) is trivial.
`.is_isomorphic` now compares the set of finite ramified places and,
unless working over $\mathbb{Q}$, the list of infinite ramified places
of both algebras. The latter can be compared as lists since the real
embeddings of the number field are sorted independently of each
algebras' invariants, but the former (probably) need to be compared as
sets since the order of the list depends on the primes above the
respective invariants. The docstring of `.is_isomorphic` (as well as
some of the other docstrings) now includes an example of a non-split
quaternion algebra with trivial discriminant, namely the algebra with
invariants $(-1,-1)$ over the quadratic field $\mathbb{Q}(\sqrt{5})$.

Possible future work:
- Extend functionality to all global fields (of characteristic not equal
to $2$) [UPDATE: Will be done once both this PR and sagemath#37554 have been
merged]
    
URL: sagemath#37173
Reported by: Sebastian A. Spindler
Reviewer(s): AurelPage, Frédéric Chapoton, grhkm21, Matthias Köppe, Sebastian A. Spindler