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https://github.com/math-comp/abel

A proof of Abel-Ruffini theorem.
https://github.com/math-comp/abel

abel-ruffini coq galois-theory mathcomp ssreflect

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A proof of Abel-Ruffini theorem.

Host: GitHub
URL: https://github.com/math-comp/abel
Owner: math-comp
Created: 2019-01-21T12:23:48.000Z (about 6 years ago)
Default Branch: master
Last Pushed: 2024-11-15T18:48:55.000Z (3 months ago)
Last Synced: 2024-11-15T19:34:42.080Z (3 months ago)
Topics: abel-ruffini, coq, galois-theory, mathcomp, ssreflect
Language: Coq
Homepage:
Size: 343 KB
Stars: 28
Watchers: 8
Forks: 8
Open Issues: 10
Metadata Files:
- Readme: README.md

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README

# Abel - Ruffini Theorem as a Mathematical Component

[![Docker CI][docker-action-shield]][docker-action-link]

[docker-action-shield]: https://github.com/math-comp/abel/actions/workflows/docker-action.yml/badge.svg?branch=master
[docker-action-link]: https://github.com/math-comp/abel/actions/workflows/docker-action.yml

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](CeCILL-B)
- Compatible Coq versions: Coq 8.10 to 8.16
- Additional dependencies:
- [MathComp ssreflect 2.0 and later](https://math-comp.github.io)
- [MathComp fingroup](https://math-comp.github.io)
- [MathComp algebra](https://math-comp.github.io)
- [MathComp solvable](https://math-comp.github.io)
- [MathComp field](https://math-comp.github.io)
- [MathComp real closed >= 2.0.0](https://github.com/math-comp/real-closed)
- Coq namespace: `Abel`
- Related publication(s):
- [Unsolvability of the Quintic Formalized in Dependent Type Theory
](https://hal.inria.fr/hal-03136002)

## Building and installation instructions

The easiest way to install the latest released version of Abel - Ruffini Theorem as a Mathematical Component
is via [OPAM](https://opam.ocaml.org/doc/Install.html):

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

To instead build and install manually, do:

``` shell
git clone https://github.com/math-comp/abel.git
cd abel
make # or make -j
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
wrapper`NullCharType` 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,

## Development information

[Developping with nix](NIX.md)