A Simple Trigonometric Classification of Quintic Roots

Abstract

This article provides a simple trigonometric method for determining how many roots of a quintic equation are real and how many are complex, without solving the equation. The approach transforms a depressed quintic $t^5+m{t}^3+n{t}^2+pt+q=0$ with $m<0$ into the trigonometric equation $f(\hat{I}¸)=\hat{I}\pm {\cos }^2\hat{I}¸+{\hat{I}}^2\cos \hat{I}¸+\cos 5\hat{I}¸+{\hat{I}}^3=0$ via the Chebyshev identity $16{\cos }^5\hat{I}¸-20{\cos }^3\hat{I}¸+5\cos \hat{I}¸=\cos 5\hat{I}¸$. The derivation is computationally light and conceptually natural, extending the quartic case to fifth-degree equations. As the Abel--Ruffini theorem forbids a general algebraic solution for the quintic, having a simple trigonometric criterion for the nature of its roots is especially appealing.

Preliminary draft (working paper). Feedback welcome; may contain errors