Évariste Galois: Principles and applications
After Abel had given a generally accepted proof of the fact that, in general, quintic polynomials cannot be solved using radicals, Galois found necessary and sufficient conditions for solvability; over the years, his theory evolved into what is now known as Galois theory. There are, however, some basic differences between the modern version and that of Galois: modern Galois theory deals with certain ring homomorphisms between field extensions, whereas the main objects in Galois's work are polynomials and certain permutations (or substitutions, as Galois called them) of their roots. Of course Galois's main results can all be formulated within the modern language, and although today we are in a better position to understand Galois's papers than his contemporaries were, working through them still requires a lot of effort, maybe more than the average working mathematician interested in the history of mathematics is willing to invest. For these mathematicians, articles such as the one reviewed here are most welcome. The article gives what may be called a `faithful representation' of Galois's article from 1831 in modern language. The original notions are explained, and then the main steps in Galois's work are discussed one by one. The last section gives applications of Galois's theory to Jacobi's modular equation, applications that Galois hinted at in his farewell letter to Chevalier.
scientific article; zbMATH DE number 1764728