This book should change, and greatly improve, our understanding of Galois’s work. That might seem an impossible claim, given the paradigm status of Galois as an abstract thinker, the bicentenary of whose birth was marked by a week-long Colloquium at the Institut Henri Poincaré in Paris in October 2011, and the many versions of Galois theory that have become central to different aspects of mathematics. But in fact the great success of Galois Theory has come to obscure the work of Galois himself, and his romantic fate has cast another layer of imagery over the historical record. This book offers the chance for us to return to the original writings.
It opens with a modest historical apparatus, where Peter Neumann discusses the possible influences on Galois, the state of the texts, previous attempts to convey some or all of what Galois wrote (from Liouville’s to the edition of Bourgne and Azra) and some of the elusive words Galois used whose meanings may have changed over the years. Then we get Galois’s papers, all in parallel form (French on the left and an English translation on the facing page), lightly but helpfully edited where necessary. First we get the five published papers, then the letter to Auguste Chevalier, the two famous memoirs that are the seed texts for Galois Theory, and finally a selection of nineteen minor mathematical manuscripts. There are also a number of illuminating reproductions of manuscript pages.
To state the obvious, this book puts what Galois wrote back in print, and much of it into English for the first time. The letter and the two memoirs are famously difficult, and Neumann supplies notes that open up some of the issues, especially the reception of the memoirs. Poisson, as he observes, has been criticised by finding the first memoir obscure and in need of a complete re-write, and Liouville was also if not defeated by it then unable to provide it with the commentary he had hoped to write. Neumann gently speculates that many of us would have done the same, and asked the brilliant but maddening author to try again. But by not responding to the memoirs by providing a modernised, rigorous account (that would surely be even longer than the originals) Neumann directs us back to the texts themselves.
The first memoir is Galois’s account of the conditions for which an irreducible polynomial equation is solvable by radicals. In it Galois took up themes from Lagrange and Gauss and pursued the idea of the ‘group of permutations’ associated to an equation. He indicated how to analyse the behaviour of this ‘group’ as the roots of auxiliary equations are adjoined, and among the cases he dealt with explicitly are the quartic equation, long known to be solvable by radicals, and irreducible equations of prime degree that are solvable by radicals, in particular such equations of degree 5. His conclusion in this case is that an irreducible quintic is solvable by radicals if and only if its group is either a certain group of order 20 that he described explicitly or a subgroup of that group.
The second memoir is on primitive equations that are solvable by radicals. For this the reader is here referred to Neumann’s earlier exploration, Neumann (2006). To my eyes the influence of Abel seems likely, but Galois did not mention his name. The letter to Chevalier is also full of remarkable insights, such as those to do with the reduction of the modular equation and the three groups we now call PSL(2, p) for p = 5, 7, 11; it is also the source of the most explicit reference Galois made to the importance of normal subgroups.
The book closes with Neumann’s reflections under the headings of myths, mysteries, and last words. Among the myths, that Galois proved that the general quintic equation is not solvable by radicals because the group A5 is simple, among the mysteries the circumstances surrounding the writing of the memoirs and why Galois did not write his memoirs down in a better form at some stage in the last two years of his life.
The last words are that much more is to be done, and here the reader may turn to the almost simultaneous appearance of a special issue of the French journal the Revue d’histoire des mathématiques. Volume 17.2 has three long essays of immediate relevance: Caroline Erhardt and Frédéric Brechenmacher write in different ways about the long passage from 1832 to a modern consensus by 1914 about what constitutes Galois Theory. There are many significant changes to the key concepts and to the place of the theory vis-à-vis other theories, and distinctively French and German versions, all of which are far from being a simple process of the accretion of results. The third essay, by Catherine Goldstein, illustrates this general message with a specific account of how Hermite responded to Galois’s ideas in his own work.
It remains to say that this edition of the mathematical writings of Évariste Galois is handsomely produced, and should become the standard reference for anyone interested in what Galois wrote.
Jeremy Gray is a Professor of the History of Mathematics at the Open University, and an Honorary Professor at the University of Warwick, where he lectures on the history of mathematics. In 2009 he was awarded the Albert Leon Whiteman Memorial Prize by the American Mathematical Society for his work in the history of mathematics. His most recent book is Plato’s Ghost: The Modernist Transformation of Mathematics, Princeton University Press, Princeton 2008, and he is presently finishing a scientific biography of Henri Poincaré.