Abel and the Theory of Algebraic Equations

Abstract

AbstractA letter from Abel to Crelle dated 25 September 1828—uncovered by Mittag-Leffler more than seventy years later—is a poignant document which throws considerable light upon the last three months that Abel had left for doing mathematics before tuberculosis incapacitated him, sapping all his strength, and eventually killing him. The letter raises the intriguing question of how the theory of equations (later known as Galois Theory) would have evolved if Abel had been able to write up his discoveries in this area. With the Abel letter as resonant backdrop we reflect upon this, basing our reflections on Abel’s published as well as posthumous manuscripts. Of vital importance is his discoveries in the theory of elliptic functions. The latter provided him with an abudance of algebraic equations.KeywordsElliptic FunctionGalois GroupGalois TheoryPrimitive ElementSplitting FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.