Galois' work beyond Galois theory on algebraic equations

Galois is renowned for his Galois theory of algebraic equations, the creation of group theory, and the dramatic story of his testamentary letter before his fatal duel. Yet, he also outlined an extraordinary and influential theory of ambiguity that deserves attention.
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Most people have heard of the sad and dramatic story of Galois and his famous testamentary letter. However, few have actually read the letter, and even fewer have read it carefully. Many assume that it primarily contains announcements of his Galois theory for algebraic equations and the related group theory. Yet, even a quick glance reveals that it holds much more, and a careful reading unveils surprises.

When I finally decided to read the entire letter, I was shocked and puzzled by his brief statement and description of a theory of ambiguity. This prompted me to explore it further--how others have interpreted it and what Galois may have intended. Understanding it was no easy task. This is not surprising, given Weyl’s famous remarks: it is "a book with seven seals," and "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."

After five years of reading and reflection, I have finally gained a deeper understanding, correcting various misunderstandings and filling in gaps. The results were recently published in a paper titled Galois's Theory of Ambiguity and Its Impacts in the Archive for History of Exact Sciences, 79, 5 (2025), pp. 1--129. https://doi.org/10.1007/s00407-024-00341-5.

One outcome of this work is an expansion of Galois’ already remarkable legacy beyond algebraic equations and group theory.

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