paulmarome / abel

A proof of Abel-Ruffini theorem.

Abel - Ruffini Theorem as a Mathematical Component

This repository contains a proof of Abel - Galois Theorem (equivalence between being solvable by radicals and having a solvable Galois group) and Abel - Ruffini Theorem (unsolvability of quintic equations) in the Coq proof-assistant and using the Mathematical Components library.

Meta

• Author(s):
  • Sophie Bernard (initial)
  • Cyril Cohen (initial)
  • Assia Mahboubi (initial)
  • Pierre-Yves Strub (initial)
• License: CeCILL-B
• Compatible Coq versions: Coq 8.10 to 8.13
• Additional dependencies:
  • MathComp ssreflect 1.11 and 1.12
  • MathComp fingroup 1.11 and 1.12
  • MathComp algebra 1.11 and 1.12
  • MathComp solvable 1.11 and 1.12
  • MathComp field 1.11 and 1.12
  • MathComp real closed >= 1.11.1
• Coq namespace: Abel
• Related publication(s): none

Building and installation instructions

The easiest way to install the latest released version of Abel - Ruffini Theorem as a Mathematical Component is via OPAM:

opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-mathcomp-abel

To instead build and install manually, do:

git clone https://github.com/math-comp/abel.git
cd abel
make   # or make -j <number-of-cores-on-your-machine> 
make install

Organization of the code

• abel.v itself contains the main theorems:

  • galois_solvable_by_radical (requires explicit roots of unity),
  • ext_solvable_by_radical (equivalent, and still requires roots of unity),
  • radical_solvable_ext (no mention of roots of unity),
  • AbelGalois, (equivalence obtained from the above two, requires roots of unity), and consequences on solvability of polynomial
  • and their consequence on the example polynomial X⁵ -4X + 2: example_not_solvable_by_radicals,

• xmathcomp/various.v contains various (rather straightforward) extensions that should be added to various mathcomp packages asap with potential minor modifications,

• xmathcomp/char0.v contains 0 characteristic specific results, that could use a refactoring for a smoother integration with mathcomp. e.g. ratr could get a canonical structure or rmorphism when the target field is a lmodType ratr, and we could provide a wrapperNullCharType akin to PrimeCharType (from finfield.v),

• xmathcomp/cyclotomic.v contains complementary results about cyclotomic polynomials,

• xmathcomp/map_gal.v contains complementary results about galois groups and galois extensions, including various isomorphisms, minimal galois extensions, solvable extensions, and mapping galois groups and galois extensions from a splitting field to another. This last construction is essential in switching to fields with roots of unity when we do not have them yet,

• xmathcomp/classic_ext.v contains the theory of classic extensions by arbitrary polynomials, most of the results there are in the classically monad, making the results available either for a boolean goal or a classical goal. This was instrumental in eliminating references to some embarrassing roots of the unity.

• xmathcomp/algR.v contains a proof that the real subset of algC (isomorphic to {x : algC | x \is Num.real}) is a real closed field (and archimedean), and endows this type algR with appropriate canonical instances.

• xmathcomp/real_closed_ext.v contains some missing lemmas from the library math-comp/real_closed, in particular bounding the number of real roots of a polynomial by one plus the number of real roots of its derivative,

• xmathcomp/diag.v contains the theory of diagonalisation and codiagonalisation with the standard criterions.

• xmathcomp/mxgal.v represents elements of a Galois group as matrices, this enables the use of diag to find eigenvectors and eigenvalues.

• xmathcomp/mxextra.v is a correspondance between 'M_(_,_) and 'Hom(_, _) which is unused in the current development, we use mxgal instead.

Development information

Developping with nix