Tuesday, January 16, 2018

Unsolvability of Quintic Equations

General formulas for roots of quadratic, cubic, and quartic equations can be written in closed form using the following algebraic operations: addition, subtraction, multiplication, division, raising to an integer power, and taking an integer root.

However, roots of quintic equations, \[ax^5 + bx^4 + cx^3 + dx^2 + e x + f = 0,\] cannot be written in closed form using these operations.

My PhD advisor, Ron Larson, had told me that this was one of the questions he was asked on his oral PhD qualifying exam. I knew the fact, but never understood the proof, since it involved math that I was not familiar with.

Fred Akalin presents a nice proof using plenty of interactive demos, visualizations, and not much advanced math.