feat(Algebra/QuadraticAlgebra): classify quadratic algebras by their discriminant

[xroblot]

Classify QuadraticAlgebra R a b up to R-algebra isomorphism by the discriminant: when 2
is invertible they are isomorphic iff the discriminants agree up to a square unit, and over ℤ
the discriminant is a complete invariant. Also adds the standard-form isomorphism
QuadraticAlgebra R a b ≃ₐ[R] QuadraticAlgebra R (discr a b) 0.

Prepared with Claude Code 🤖

---

• depends on: [Merged by Bors] - feat(Algebra/QuadraticAlgebra): discriminant and field criterion #42206
• depends on: chore(Algebra/QuadraticAlgebra): rename map/mapEquiv to changeGenerator #42751

[github-actions]

PR summary 383452c168

Import changes exceeding 2%

%	File
+22.64%	Mathlib.Algebra.QuadraticAlgebra.Discr

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Algebra.QuadraticAlgebra.Discr	1365	1674	+309 (+22.64%)

Import changes for all files

Files	Import difference
Mathlib.Algebra.QuadraticAlgebra.Discr	309

---

Declarations diff (regex)

+ algEquivDiscrZero
+ basis_apply_one
+ basis_apply_zero
+ discr_eq_im_sq_mul_discr
+ discr_eq_im_sq_mul_discr'
+ im_sq_mul_discr
+ isUnit_im_omega_of_algEquiv
+ map_star
+ nonempty_algEquiv_iff
+ nonempty_algEquiv_iff_of_invertible_two
+ nonempty_algEquiv_int_iff
+ norm_map_omega
+ smul_omega_sub_eq
+ trace_map
+ trace_map_omega

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 383452c).

• +14 new declarations
• −0 removed declarations

+QuadraticAlgebra.algEquivDiscrZero
+QuadraticAlgebra.basis_apply_one
+QuadraticAlgebra.basis_apply_zero
+QuadraticAlgebra.discr_eq_im_sq_mul_discr
+QuadraticAlgebra.discr_eq_im_sq_mul_discr'
+QuadraticAlgebra.im_sq_mul_discr
+QuadraticAlgebra.isUnit_im_omega_of_algEquiv
+QuadraticAlgebra.map_star
+QuadraticAlgebra.nonempty_algEquiv_iff
+QuadraticAlgebra.nonempty_algEquiv_iff_of_invertible_two
+QuadraticAlgebra.nonempty_algEquiv_int_iff
+QuadraticAlgebra.norm_map_omega
+QuadraticAlgebra.trace_map
+QuadraticAlgebra.trace_map_omega

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 383452c168
Reference commit caa4dbde35

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).

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - feat(Algebra/QuadraticAlgebra): discriminant and field criterion #42206
• chore(Algebra/QuadraticAlgebra): rename map/mapEquiv to changeGenerator #42751
  By Dependent Issues (🤖). Happy coding!