  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 $+,-,\times,\div,$ 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 four or less (and how they are constructed), and why they
do not for degree five or more.

History and Overview;Algebraic Geometry