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gh-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 #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 #37554 have been
merged]
URL: #37173
Reported by: Sebastian A. Spindler
Reviewer(s): AurelPage, Frédéric Chapoton, grhkm21, Matthias Köppe, Sebastian A. Spindler
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