Galois for 21st-Century Readers

T he recent bilingual publication of The Mathematical Writings of Évariste Galois by Peter M. Neumann [6] will make Galois’s own words available to a vast new audience of students of modern algebra. I have long advocated reading the original works of great mathematicians, but even with the advantage of Neumann’s extensively annotated transcription and translation it will be difficult for modern readers to connect Galois theory as they know it with Galois’s original presentation of it in his famous First Memoir (Premier Mémoire), entitled “On the conditions for the solvability of equations by radicals”. The First Memoir was submitted to the Paris Academy of Sciences in January of 1831, only to be rejected. With the benefit of hindsight, it is easy to condemn this rejection as an epicmisjudgment. However, anyone who has studied the memoir will sympathize with the decision, especially in view of the fact that the referees recommended that the young author—Galois was just nineteen at the time—make his presentation clearer and more expansive. They could not have imagined that this would be their last chance to recognize the merit of the work of an unparalleled genius. In this paper, I have tried to explain the First Memoir to modern readers, going through it proposition by proposition. The most important proposition, and the one I most emphasize, is Proposition 2, the one about which Galois wrote in the margin, “There is something to be completed in this proof. I do not have the time” (the sections “Proposition 2” and “Proposed Revision of