A conjecture on the forms of the roots of equations

TL;DR

Euler explores the possibility of solving polynomial equations by constructing resolvent equations of lower degree, analyzing their roots to find solutions for quadratic, cubic, quartic, and higher-degree equations, and investigates conditions for their effectiveness.

Contribution

The paper investigates Euler's approach to solving equations via resolvent equations, providing insights into the method's applicability to various degrees and proposing new techniques for quartic solutions.

Findings

Resolved quadratic, cubic, and quartic equations using resolvent methods.

Identified cases where resolvent equations effectively solve higher-degree equations.

Proposed a new method for solving quartic equations.

Abstract

E30 in the Enestrom index. Translated from the Latin original "De formis radicum aequationum cuiusque ordinis coniectatio" (1733). For an equation of degree n, Euler wants to define a "resolvent equation" of degree n-1 whose roots are related to the roots of the original equation. Thus by solving the resolvent we can solve the original equation. In sections 2 to 7 he works this out for quadratic, cubic and biquadratic equations. Apparently he gives a new method for solving the quartic in section 5. Then in section 8 Euler says that he wants to try the same approach for solving the quintic equation and general nth degree equations. In the rest of the paper Euler tries to figure out in what cases resolvents will work. Two references I found useful were Chapter 14, p.p. 106-113 of C. Edward Sandifer, "The Early Mathematics of Leonhard Euler", published 2007 by The Mathematical Association of America and Olaf Neumann, "Cyclotomy: from Euler through Vandermonde to Gauss", p.p. 323-362 in the collection "Leonhard Euler: Life, Work and Legacy" edited by Bradley and Sandifer, 2007. Stacy Langton has given a lot of details about Euler's work on the theory of equations, and also some advice on the translation; of course any mistakes are my own. If Langton ends up writing anything about Euler' and the theory of equations I would highly recommend reading it.