I am happy to announce that we -- Sophie Bernard, Cyril Cohen, Assia Mahboubi and Pierre-Yves Strub -- were able to formalize in Coq **a proof of Abel-Ruffini Theorem**, which states that there are polynomials of degree 5 that are not solvable by radicals, e.g. $X^5 - 4X + 2$.

```
Lemma example_not_solvable_by_radicals :
~ solvable_by_radical_poly ('X^5 - 4 *: 'X + 2).
```

This is a consequence of Abel-Galois theorem (also formalized) which states the equivalence between being solvable by radicals and having a solvable Galois group.

The proofs are accessible in the repository https://github.com/math-comp/Abel and will soon be released as the opam package `coq-mathcomp-abel.1.0.0`

and as a nix package. This development uses and extends non trivially the Mathematical Components library especially the Galois Theory part.

NB: all the proofs in this repository are constructive.

–

Cyril Cohen, for the contributors of https://github.com/math-comp/Abel

Cross posted to #Coq users and #math-comp users please follow-up here.

so is this the first full formalization of Abel-Ruffini? I guess the recent formalization work presented at Lean Together is quite different conceptually? Or how do they compare?

by the way, I think this deserves a CoqLang tweet @Anton Trunov @Emilio Jesús Gallego Arias

The work presented at the lean together workshop was a road map towards Abel-Ruffini, they are not there yet.

Yep, was typing the tweet this instant :)

Cyril Cohen said:

The work presented at the lean together workshop was a road map towards Abel-Ruffini, they are not there yet.

I suspected as much, but they also aim to prove it in a quite different way, right?

It's the same proof, although the tools and fundations are not exactly the same, they should look alike

ah OK, then I might've misunderstood something in the talk.

Can you do something to make `2%:Q%:P`

look less ugly?

I mean, for the sake of the example, I would not mind locally adding ad hoc notations for numbers

Anyway, chapeau !

Ahah, right, I could do that... but I'm done for today

Actually I think we should try to make numerals work in the ring scope now...

Enrico Tassi said:

Can you do something to make

`2%:Q%:P`

look less ugly?

I took care of this!

Once the lean proof is finished it would be quite an interesting point of comparison.

“Oh, l’Abel-preuve !”

Last updated: Feb 08 2023 at 04:04 UTC