feat(Algebra/Quaternion): quaternion are central simple

[Mathias-Stout]

We show that the quaternion algebra ℍ[R,a,b,c] over a field R is a central simple R-algebra, provided that c * (b ^ 2 + 4 * a) ≠ 0.

• depends on: chore(Algebra): make quaternion directory #41536
• depends on: feat(RingTheory/TwoSidedIdeal/Lattice): add lemmas on two-sided ideals #41537

Co-authored-by: Justus Springer justusspringer@gmx.de

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[github-actions]

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[github-actions]

PR summary 86493005d2

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Algebra.QuaternionBasis	-1346
Mathlib.Algebra.Quaternion	-1330
Mathlib.Algebra.Quaternion.Basic	1330
Mathlib.Algebra.Quaternion.Basis	1346
Mathlib.Algebra.Quaternion.CentralSimple (new file)	1347

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Declarations diff (regex)

+ eq_top_of_isUnit_mem
+ eq_top_of_mem_imJ_imK
+ eq_top_of_mem_re
+ exists_mem_ne_zero_of_ne_bot
+ isCentral
+ isSimpleRing
+ isUnit_iff
+ isUnit_of_isUnit_re
+ ne_bot_iff

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 8649300).

• +7 new declarations
• −0 removed declarations

+QuaternionAlgebra.isCentral
+QuaternionAlgebra.isSimpleRing
+QuaternionAlgebra.isUnit_iff
+QuaternionAlgebra.isUnit_of_isUnit_re
+TwoSidedIdeal.eq_top_of_isUnit_mem
+TwoSidedIdeal.exists_mem_ne_zero_of_ne_bot
+TwoSidedIdeal.ne_bot_iff

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5005	1	exposed public sections

Current commit 86493005d2
Reference commit 2d00c9242a

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

---

note: file Mathlib/Algebra/QuaternionBasis.lean` was renamed to `Mathlib/Algebra/Quaternion/Basis.lean without a module deprecation
Please create a follow-up pull request adding one. Thanks!

note: file Mathlib/Algebra/Quaternion.lean` was renamed to `Mathlib/Algebra/Quaternion/Basic.lean without a module deprecation
Please create a follow-up pull request adding one. Thanks!

[mathlib-dependent-issues]

This PR/issue depends on:

• chore(Algebra): make quaternion directory #41536
• feat(RingTheory/TwoSidedIdeal/Lattice): add lemmas on two-sided ideals #41537
  By Dependent Issues (🤖). Happy coding!