Dear Coq community,
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 GitHub - math-comp/Abel: A proof of Abel-Ruffini theorem. 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.
Happy new year and best wishes,
Cyril Cohen, for the contributors of GitHub - math-comp/Abel: A proof of Abel-Ruffini theorem.