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Algebra: Polynomials, Galois Theory and Applications

Frédéric Butin
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Dover Publications is generally known in the mathematical community for reissuing, and saving from extinction, classic texts from yesteryear. It has recently expanded its horizons, however, and now offers the Aurora: Dover Modern Math Originals series of original texts, of which the book now under review is perhaps the most recent example. Actually this book is not really new; it is a translation of a book originally published in French in 2011.

The focus of the book is on fields and Galois theory and applications of these subjects. As befits its European origins, it is a demanding text, rather concisely written and not given to a great deal of “hand-holding”. (As an example, the proof that the set of algebraic numbers is countable is given in less than four lines of text.) It presupposes completion of a good first course in abstract algebra that covered groups, rings and fields. While some topics in group theory are quickly reviewed, the selection of what topics to review and what to take for granted occasionally seems somewhat ad hoc; the definition of a group is assumed, for example, but the author does define what it means for an element of a group to have finite order. Then, in the next paragraph, he assumes as known the notion of a group acting on a set, including terms like “orbit”. The rudiments of basic number theory (including congruence modulo \(n\)) are also taken for granted, as are the definitions of “ring” and “field” and the basic facts about these objects. It is also assumed that the reader knows what Zorn’s Lemma is; on page 24, for example, the author reminds the reader (without proof) that any proper ideal in a commutative ring with identity is contained in a maximal one, and in a footnote states “As one can imagine, Krull’s theorem is based on Zorn’s Lemma!”.

Though this is a fairly slim book (about 250 pages of actual text), it covers a lot of interesting mathematics, including (as we’ll see below) a number of topics and results that are not easily found in the textbook literature.

There are three parts to the book, each one consisting of several chapters. Part I, consisting of two chapters, is entitled “Arithmetic, Rings and Polynomials”, but there’s much more here than this title would suggest. The reader will find in these chapters a brief introduction to some topics in cryptography (RSA, ElGamal) and also a much deeper look at polynomials than one typically encounters, including concepts like the resultant and the permanent. In an unusual feature, it is proved that the “generic” determinant, resultant and permanent polynomials are irreducible. The symmetric and alternating groups are also considered in some detail, including a proof of the simplicity of the alternating groups \(A_n\) for \(n>4\). (The author uses the more cumbersome Fraktur notation \(\mathfrak{A}_n\), however.)

Here, and elsewhere in the text, the author makes use of Maple software, but he notes in the preface that other software could also be used by the reader if desired.

The five chapters that constitute Part II of the text address the theory of field extensions and Galois theory. The material is developed in a fairly general form; some introductory texts, for example, limit themselves to the characteristic \(0\) situation (thereby avoiding the problem of dealing with non-separable extensions) but this text does not opt for that simplification. Here again, we also find interesting results that are not often found in the literature. The existence of an algebraic closure of an arbitrary field is proved, formulas of Cardano and Ferrari for the roots of cubic and quartic polynomials are addressed in some depth, and there is a statement and proof of a theorem of Van der Waerden for computing Galois groups of polynomials.

Some of the terminology and notation here may seem, to an American reader, to be a little strange; for example, the author uses the phrase “rupture fields”, which I have never heard before, to mean what I would call “simple algebraic extensions” of a field, and he denotes such an extension by \(F(x)\), using \(x\) to denote the root of a polynomial in an extension field. To my mind, however, \(x\) should always denote a transcendental element and \(F(x)\) should denote the field of rational functions in \(x\). Readers who, like me, have been brought up on this convention may find the notation a bit jarring.

Part III of the text discusses applications of the foregoing material to such diverse areas as compass and straightedge constructions or finite fields and error-correcting codes. A final chapter introduces some basic ideas of algebraic number theory. Once again we find things here that are not easy to find elsewhere. The theorem that characterizes those positive integers \(n\) for which a regular \(n\)-gon is constructible is not only stated (which is fairly common) but proved (which is not). In addition, error-correcting codes are discussed in considerable detail, and the book also treats Berlekamp subalgebras and the Berlekamp algorithm.

Each chapter ends with exercises. Few of them are trivial, and some of them are quite challenging. (One asks the reader, for example, to prove the familiar characterization of even perfect numbers.) Each exercise, however, is followed immediately by its solution. I know that opinions differ as to the advisability of providing solutions to students, but I generally come down against it, particularly where, as here, the student doesn’t even have to go to the trouble of flipping to the back of the book to read the solution. The value of an exercise lies in having the student think about it for awhile, and I think it’s asking an awful lot of students to tell them to refrain from reading a solution that comes right on the heels of the problem itself.

There is also a six-page summary of capsule biographies of various mathematicians at the end of the text. These biographical summaries only run to a few lines each and do not do more than give the years of the subject’s birth and death, and summarize quickly one or two major accomplishments of the person involved. Galois’ account, for example, is three lines long and does not even mention the duel that ended his life.

To summarize: an interesting book to have on one’s shelf, but there are already several very good books on Galois theory available (the ones by David Cox and Ian Stewart spring immediately to mind) and, given the density of this text, I don’t see this book as a pedagogical advance on them.

Mark Hunacek ( teaches mathematics at Iowa State University. 

The table of contents is not available.