The Abel–Ruffini Theorem: Complex but Not Complicated
Ramond, Paul (2022), The Abel–Ruffini Theorem: Complex but Not Complicated, The American Mathematical Monthly, 129, 3, p. 231-245. 10.1080/00029890.2022.2010494
TypeArticle accepté pour publication ou publié
Journal nameThe American Mathematical Monthly
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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 +,−,×,÷, and √. 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.
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