Unsolvability of the Quintic Formalized in Dependent Type Theory

Authors Sophie Bernard, Cyril Cohen , Assia Mahboubi, Pierre-Yves Strub

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Author Details

Sophie Bernard
  • Université Côte d'Azur, Inria, Sophia Antipolis, France
Cyril Cohen
  • Université Côte d'Azur, Inria, Sophia Antipolis, France
Assia Mahboubi
  • Inria, Nantes, France
  • Vrije Universiteit Amsterdam, The Netherlands
Pierre-Yves Strub
  • École Polytechnique, Palaiseau, France

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Sophie Bernard, Cyril Cohen, Assia Mahboubi, and Pierre-Yves Strub. Unsolvability of the Quintic Formalized in Dependent Type Theory. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this paper, we describe an axiom-free Coq formalization that there does not exists a general method for solving by radicals polynomial equations of degree greater than 4. This development includes a proof of Galois' Theorem of the equivalence between solvable extensions and extensions solvable by radicals. The unsolvability of the general quintic follows from applying this theorem to a well chosen polynomial with unsolvable Galois group.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Logic and verification
  • Theory of computation → Constructive mathematics
  • Galois theory
  • Coq
  • Mathematical Components
  • Dependent Type Theory
  • Abel-Ruffini
  • General quintic


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