[Merged by Bors] - feat: Polynomial FLT

[seewoo5]

Prove the polynomial FLT, using Mason-Stothers theorem #15706. More generally, prove non-solvability of Fermat-Catalan equation: $ux^p + vx^q + wz^r = 0$ where $u, v, w \in k^\times$ are units and $p, q, r \ge 3$ with $pq + qr + rp \le pqr$.

Also derive the IsCoprime b c and IsCoprime c a assumptions from the IsCoprime a b and a + b + c = 0 ones in Polynomial.abc.

---

• depends on: [Merged by Bors] - feat:Polynomial.derivative_C_mul #19048
• depends on: [Merged by Bors] - feat: isCoprime_mul_units_left, isCoprime_mul_units_right #19133
• depends on: [Merged by Bors] - chore(Radical): minor refactoring of lemmas #19875
• depends on: [Merged by Bors] - feat: Two lemmas on divisibility and coprimality of expand #20143
• depends on: [Merged by Bors] - feat(Polynomial): (a^n)' = 0 iff a' = 0 when n doesn't divide the characteristic #20190
• depends on: [Merged by Bors] - feat(GroupWithZero): mul_(eq/ne)_zero_iff_(left/right) #20191
• depends on: [Merged by Bors] - feat(Radical): (radical a).natDegree ≤ a.natDegree #20468

[github-actions]

PR summary 31b1f512b3

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.FLT.Polynomial (new file)	1190

---

Declarations diff

+ Polynomial.flt
+ Polynomial.flt_catalan
+ fermatLastTheoremWith'_polynomial

You can run this locally as follows

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

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[seewoo5]

@jcpaik Could you help me to remove sorrys? Also, flt_catalan_aux haven't fixed yet.

[jcpaik]

@jcpaik Could you help me to remove sorrys? Also, flt_catalan_aux haven't fixed yet.

Merge the PRs then the file is sorry-free and proves polynomial FLT. We only then need to golf/separate the file into smaller pieces.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat:Polynomial.derivative_C_mul #19048
• [Merged by Bors] - feat: isCoprime_mul_units_left, isCoprime_mul_units_right #19133
• [Merged by Bors] - chore(Radical): minor refactoring of lemmas #19875
• [Merged by Bors] - feat: Two lemmas on divisibility and coprimality of expand #20143
• [Merged by Bors] - feat(Polynomial): (a^n)' = 0 iff a' = 0 when n doesn't divide the characteristic #20190
• [Merged by Bors] - feat(GroupWithZero): mul_(eq/ne)_zero_iff_(left/right) #20191
• [Merged by Bors] - feat(Radical): (radical a).natDegree ≤ a.natDegree #20468
  By Dependent Issues (🤖). Happy coding!

[YaelDillies]

Looks good to me! @kbuzzard ?

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by YaelDillies.

[kbuzzard]

A couple more docstring nits. Thanks a lot!

bors d+

[mathlib-bors]

✌️ seewoo5 can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

[seewoo5]

Thank you all!

bors r+

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• Build
• Lint style