Felix Nordgren Odhner: Polynomials of Degree 3 and 4: Classical Solution Methods and Their Significance

Abstract: This thesis examines classical solution methods for third- and fourth-degree polynomial equations, developed by Cardano and Ferrari in the sixteenth century. Using modern algebraic notation, these methods are derived step by step and framed in terms of polynomial rings, symmetric polynomials, and Viète’s relations.

The work emphasizes the underlying algebraic structure of the methods, showing how specific substitutions systematically eliminate terms and reduce the equations to simpler forms. In particular, Cardano’s solution of the depressed cubic is shown to form the foundation of Ferrari’s method for solving quartic equations via a cubic resolvent.

Finally, the thesis discusses the theoretical limitations of these classical approaches. Results by Ruffini, Abel, and Galois demonstrate that no general formula exists for polynomial equations of degree five or higher, as the symmetries of their roots cannot, in general, be resolved using radicals. The thesis thus illustrates both the historical development of algebraic solution methods and the structural reasons for their limitations.