Galois’s theory of ambiguity and its impacts

Abstract

Although many people have extensively studied the earlier parts of Galois’s testamentary letter, in particular those concerning the Galois theory of algebraic equations and related group theory, it seems that the theory of ambiguity near the end of his letter is less well known and studied, and therefore, remaining somewhat mysterious. One purpose of this paper is to provide an overview of diverse interpretations of Galois’s theory of ambiguity by people such as Lie, Klein, Picard, and Grothendieck. We will discuss how well they fit Galois’s description for this theory and whether they satisfy one important criterion set by him. After a careful analysis of Galois’s statements regarding the theory of ambiguity and the rationale behind them, by taking all Galois’s works into consideration, we will offer our interpretation of it through the theory of monodromy for linear differential equations. Our findings challenge the common perception that Galois could not foresee applications of group theory beyond algebraic equations. Subsequently, we will discuss how these various interpretations have influenced later development of mathematics, particularly their impact on Lie’s idée fixe to develop a theory of transformation groups for differential equations. This analysis also raises doubts about a certain aspect of the commonly accepted narrative regarding the origin of the theory of Lie groups, and provides one important example of theories partially motivated by Galois’s theory of ambiguity. Additionally, we will identify results from works of his near contemporaries such as Riemann, Fuchs, Jordan and later generations such as Siegel, which seem to fit well our rendering of Galois’s description and criterion. This demonstrates the potentially intended broad scope of Galois’s theory of ambiguity. Furthermore, their alignment with our interpretation of Galois’s theory of ambiguity adds feasibility and credibility to the latter. We hope that the analysis in this paper will enhance our understanding of the meaning and impacts of Galois’s theory of ambiguity, reaffirming the profound and broad vision that Galois held for mathematics. Moreover, this paper contributes to an effort to reevaluate some of Galois’s seminal contributions and their impacts on the development of mathematics, transcending the traditional boundaries of algebra and number theory.