Skip to content

Commit 42d00ed

Browse files
author
Release Manager
committed
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
2 parents 8d7107c + 5aeb5c4 commit 42d00ed

File tree

1 file changed

+376
-84
lines changed

1 file changed

+376
-84
lines changed

0 commit comments

Comments
 (0)