The Abel–Ruffini Theorem: Complex but Not Complicated

Abstract

In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel–Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using radicals, that is, using its coefficients and arithmetic operations $+,-,\times ,÷,$ and $\sqrt{}$. The present article is written purposely with concise and pedagogical terms and dedicated to students and researchers not familiar with Galois theory, or even group theory in general, which are the usual tools used to prove this remarkable theorem. In particular, the proof is self-contained and gives some insight as to why formulae exist for equations of degree less than five (and how they are constructed), and why they do not for degree five or more.

MSC:

• Primary 12D10
• Secondary 00A06; 00-01

Acknowledgments

We thank to the referees for their helpful remarks and comments that greatly improved the quality of the article. We also are indebted to Boaz Katz and Leo Goldmakher for mail exchanges on their respective works that exposed the author to this beautiful piece of mathematics.

Additional information

Notes on contributors

Paul Ramond

PAUL RAMOND is a doctoral student at Paris University, completing a Ph.D. in theoretical astrophysics at the Obervatoire de Paris. He is also interested in the history and interplay between mathematics and physics. Laboratoire Univers et Thories, CNRS, Observatoire de Paris, 92190 Meudon, France. [email protected]