An Elementary Proof of the Unsolvability of Quintic Equations by Roua Agrebi '19

An Elementary Proof of the Unsolvability of Quintic Equations by Roua Agrebi ’19, Monday, February 18, Stetson Court Classroom 101, Mathematics Colloquium

Abstract:  The solvability of polynomials has occupied the minds of mathematicians for centuries. In the 16th century solutions to cubic and quartic equations were discovered and mathematicians attempted to use the same methods to find a solution for the quintic equation. In 1771 Lagrange came up with the novel idea of relating the solvability of equations to the theory of groups of permutations. In the following decades, Ruffini and Abel prove that the quintic is unsolvable.
The Abel-Ruffini Theorem states that there is no general algebraic solution to polynomial equations of degree five or higher. The usual approach to the above claim involves a semester’s worth of abstract algebra and Galois theory. However, in 1963 Vladimir Arnold discovered a topological proof which is elementary and requires no knowledge of abstract algebra or Galois Theory. Arnold’s proof gives a stronger result than Galois Theory since it allows not only the use of radicals but also of any continuous function such as sin(), exp(), etc.

This talk will explain Arnold’s proof which provides an accessible and elementary way of understanding and proving the unsolvability of the quintic. We will start by proving that there does not exist a continuous function that solves the quadratic polynomial. Then we will address the cubic case proving that any solution must contain nested radicals. Finally, we will use properties of permutations and commutators to prove the unsolvability of the quintic.