Review of Peter Neumann's book on Galois

Almost every major university in the world offers a course in Galois Theory. Most of these courses will recount briefly (and often inaccurately) the remarkable circumstances of the birth of this beautiful theory. The drama of Galois' short life and his death in a duel tend to take precedence over the revolution he created in mathematics. The monograph under review is an edition of the writings of Galois, with commentary, trying to establish both a French and an English version of his writings. Most of these writings were unpublished in his lifetime and much of Galois' manuscripts contain a jigsaw puzzle of little scraps which are hard to fit together. This book is not an attempt to make mathematical sense of these manuscripts, rather it is an outstanding attempt to present in printed form something as close as possible to what Galois committed to paper.
During his lifetime, Galois published five articles. The unpublished manuscripts have been organized into 25 dossiers and are held in the library of the Institut de France. Some additional material, such as copies of some manuscripts by Auguste Chevalier and corrected proofs by Liouville when he worked on this material over 10 years after Galois' death, are also in the dossiers. Neumann attempts to present in print as faithful a copy as possible of most of this material. Where words have been scored out or overwritten, this has been accurately transcribed. When Galois adds a remark that a particular sentence should be moved to another place in his work, those who previously presented Galois' work carried out his instructions and moved the sentence to the position he indicates. Neumann, however, leaves such a sentence where Galois wrote it, reproduced along with Galois' instruction as to where it should be moved. The original French text occupies the lefthand pages of the monograph, while the righthand pages contain an English translation. All of this material has been published before in the original French, some of it many times over. Some of it has been published before in English translation, but this book contains the first English translation of much of the material.
The Introduction contains a brief biography of Galois, a look at his mathematical background, a detailed look at previous publications of Galois' writings and a detailed look at the difficulties of translation, particularly looking at words which have changed their meanings. Of course, Galois' work contains misprints and misspellings and Neumann gives a useful list of the commonest of these errors. Note that, keeping to the philosophy of reproducing what Galois wrote as accurately as possible, these errors have been faithfully reproduced in the French text. Chapter 2 of the monograph looks at the five articles published in Galois' lifetime. The commentary here looks at differences between the original papers and later publications where errors were corrected. Chapter 3 gives the famous Testamentary Letter written to Auguste Chevalier by Galois on the night before he took part in the duel. This fantastic letter, about six and a half pages long, summarizes what Galois knew. It may, as Neumann suggests, contain more mathematics than anything written before or later of equal length. Here the commentary gives accurate measurements of the paper on which it is written and precise details of differences between various versions published later are all noted. The Testamentary Letter is a difficult document. Chapter 4 gives the First Memoir, the paper which Galois submitted for publication but which was rejected, in which he gave conditions under which an irreducible polynomial equation can be solved by radicals. Neumann gives the differences between Galois' manuscript, Chevalier's copy and various printed versions. He also gives notes including correspondence from Galois and the referees. Chapter 5 treats Galois' Second Memoir, on primitive equations that can be solved by radicals, in a similar fashion. Chapter 6, almost half of the monograph, gives dossiers 624. Some of these contain essays by Galois, probably never intended for publication, while others contain material which Galois was working on relating to material he hoped to publish. The final chapter is entitled Epilogue: myths and mysteries.
In addition to indicating what Neumann's monograph contains, it is important to indicate what it does not contain. This is clearly stated in the text: the commentary is focussed on content, not meaning; on syntax, not semantics; on relationships with previous editions. ... I have tried to suppress my mathematical instincts. While fully understanding the reasons behind this philosophy, it is still the case that at many points the reviewer regretted that Neumann had not made some enlightening mathematical comment. This monograph is an outstanding contribution to the history of mathematics. It surely will become a standard reference for anyone wishing to return to the source of Galois' writings. At many places in the work Neumann indicates that he hopes to discuss a particular point in a future article. The reviewer will surely echo the wishes of the whole mathematical community in encouraging Peter Neumann to find the time to write such articles.
JOC/EFR April 2016
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