A Short Elementary Proof of the Insolvability of the Equation of Degree 5

We present short elementary proofs of the well-known Ruffini-Abel-Galois theorems on unsolvability of algebraic equations in radicals. This proof is obtained from existing expositions by stripping away material not required for the proof (but presumably required elsewhere). In particular, we do not use the terms ‘Galois group’ and even ‘group’. However, our presentation is a good way to learn (or recall) the starting idea of Galois theory: to look at how the symmetry of a polynomial is decreased when a radical is extracted. So the note provides a bridge (by showing that there is no gap) between elementary mathematics and Galois theory. The note is accessible to students familiar with polynomials, complex numbers and permutations; so the note might be interesting easy reading for professional mathematicians.