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Galois Theory

Steven H. Weintraub
Springer Verlag
Publication Date: 
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Marion Cohen
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When I first saw this book on the huge list that MAA reviewers get to choose from, I was very excited. I’ve always wanted to re-learn Galois Theory. I’ve loved its ideas (for the same reasons that most mathematicians do, in particular for its ingenious and complicated ways of dealing with rather familiar elementary-sounding problems), but since graduate school I’ve never needed to use it in either my research or my teaching (or my poetry ), so it wasn’t as much a part of my life, and mind, as I’ve wanted it to be. In particular, I wanted to possess a more gut-level knowledge of the reason that there’s no “quintic formula”. So I googled this book to see whether it would fit one of my foremost book-reviewing criteria, and found that, yes, it’s short enough. (Wow! Galois Theory in 182 pages!) So, armed with this book (plus Ian Stewart’s book of the same title, which I already owned), I set out to work at realizing this one of my many life-dreams.

Weintraub’s book was a big help. First, he kindly devotes the entire first chapter to examples; readers don’t have to know anything at all at this stage; they can just look through these examples, along with the several “preview-type sentences” provided, and begin to get the idea. Or some idea. One could begin to suspect, e.g., that there’s some difference between “adding the square root of 2” to the rationals and adding the cube root of two. A first chapter devoted to examples is, I think, a good idea.

I almost wish that chapter were longer. I also wish that he had included exercises at the end of it. But perhaps he had his reasons. For example, longer could become overwhelming, and defeat the purpose. Exercises might do the same. Still, strategically chosen exercises possibly wouldn’t. They might invite us, e.g., to investigate “adding on” to the rationals the square root of two plus the square root of three — what would that field look like? Perhaps only one or two exercises of this kind would get students into the spirit. (I’ll grant, though, that the exercises in later chapters make up for this first-chapter deficiency, in both quantity and difficulty.)

Along these same lines, on page 127 Weintraub does something for which students will be very thankful. “For the convenience of the reader, we shall collect most of these results in this section. We shall keep the original numbering to make it easy for the reader to refer back…” What “these results” have in common is, for our purposes, their usefulness in actually computing Galois groups — an activity which, happily, concludes this section.

My only possible criticism of this book (besides the typos which I kept finding) is that I wish that it had more examples throughout. This is perhaps “just me”. But I found that, as I read, and as I learned, I needed to discover and work out examples for myself (and I needed to know whether I was doing this correctly). This was especially apparent on page 25, where Weintraub introduces normal, separable, and Galois extensions. I needed, or wanted, in a hurry, examples of both normal and non-normal extensions, and ditto separable.

After a while I thought, feeling a bit exasperated, “How can a minimal polynomial not be separable?” Fortunately, I’m a skipper-around-er, and I soon landed in the chapter where he shows that, indeed, for F = Q (the field of rationals), a minimal polynomial must be separable. In fact, perhaps that’s precisely Weintraub’s point, to get his readers to skip around. That would go along with his many passages where he quotes future theorems, and sometimes even uses future theorems in his proofs. Also, some of the exercises seemed to necessitate the use of future theorems (and this was not always stated). Since I’m a skipper-around-er, all that worked for me, and I think it would eventually also work for most non-skipper-around-ers (though perhaps they’d feel some exasperation and suspense).

This book packs a wallop in its 182 pages. After that short first chapter, it launches into a longer one on field extensions and everything else that that chapter needs in order to earn its title, namely “The Fundamental Theorem of Galois Theory”. The third chapter, “Development and Applications of Galois Theory” is just that: symmetric polynomials, elementary symmetric polynomials, and everything else symmetric, separable extensions, finite fields, disjoint extensions with all the important technical theorems and corollaries, simple extensions (for example, what simple extension we get by adjoining the square, cube, fifth, and seventh roots of 2 — although needing to forgo the complete proof ‘til a later chapter), the normal basis theorem, Abelian extensions and Kummer fields, norm and trace, In Chapter 4 we find out everything we wanted to know about Q, in particular all those ancient Greek impossibilities, along with that “impossible quintic”. The fifth and final chapter, “Further Topics in Field Theory”, contains beauties like “all” algebraic closures (in particular, C ) and the Fundamental Theorem of Infinite Galois Theory.

Like, probably, any student learning or re-learning any subject, I’ve harbored a couple of Galois hang-ups over the years. First, I have always felt, if erroneously, sort of disappointed that Galois Theory did not take the approach of directly showing that the general quintic is unsolvable, but instead shows that there are specific quintics which are unsolvable (and, yes, showing precisely which ones are and which aren’t). Yes, the latter is more, but again, my personal interest lay in knowing why there’s no “quintic formula”; I have felt that somehow specific quintics are less mysterious, less exciting, than “the general quintic”. Second, and related as it turned out, I didn’t remember much that I might have learned in graduate school about the history, and I wondered why the proof of the insolvability of the quintic seemed to be attributed to both Galois and Abel.

These vague confusions had been on my back burner for a long time, and stayed there as I read this book, until I consulted Stewart, who goes more into the history of “the quintic problem”. Indeed, Weintraub does it all in one gulp — meaning bypassing Ruffini and Abel — also all fields from the gitgo, not only Q — so gulp we must. This does have its advantages (namely, we gain insight which we might otherwise not).

Still, that, for me, was one of several glitches in the “Galois Theory in 182 pages” bargain. Omitting the history means omitting some of the perspective, as well as some of the mathematics — although opting to concentrate on Galois Theory seems to be a valid approach; it’s just different. Nonetheless, reading Weintraub’s book gives the impression that Galois Theory is absolutely necessary for the proof and understanding of the insolvability of “the general quintic”, whereas that’s not the case.

A second glitch in the bargain, for some, is that there are quite a lot of prerequisites – in particular, his previous book, which he refers to as AW (meaning Algebra: An Approach via Module Theory, by Adkins and Weintraub) and which includes group and field theory. (However, the definition of “solvable group” and the proof of the crucial fact that the symmetric group on five elements is not solvable are in an Appendix.)

Another confusion that I still harbor is this: It seemed to me that Galois Theory (in the sense of The Fundamental Theorem) is not necessary in order to deal with the three most well-known “impossibilities” (angle trisection, etc.) but only the theorems about field extensions. This seems important to the understanding of the whole picture, and I wonder why so many texts, including both Weintraub and Stewart, make no mention of this (that I could find).

In my quest for a gut-level understanding of Galois Theory, I found it best to go back and forth between the two books (rather like a kid playing its mother against its father. For me, the union of the books was important (along with the comfort of their huge intersection …). In comparing them, I’d say that there are advantages to each (and, again, I’m glad that I own both). Weintraub is concise, rather “pure”, and again, gives us insights that are less apparent when we’re given “Galois Theory” in two big gulps, as in Stewart (first Q, then F). Stewart gives us “a kinder, gentler” Galois”, with more details (and more examples… I must admit, however, that Weintraub’s one big gulp is, ultimately, no harder to swallow than Stewart’s two; F is not much more difficult to visualize than Q.

For a long time I reviewed mostly books on “math culture” or “math for non-majors”. So I was big on pedagogy, realizing that it’s important for their readerships. For the past several years, however, I’ve taken on books that are meant for more “advanced” math readers, in particular graduate or late-undergraduate students in math. For these books I tend to feel that pretty much anything goes; the readers will get it.

I do, however, care aesthetically about presentation; I care about the public image of the math itself. So for example, even though I know that readers will eventually understand a proof that neglects to mention its essence, I would prefer that that proof (and theorem) go down in history in a way that includes that essence. In other words, sometimes I’m a nitpicky “purist”. That comment does not apply very much to our book here. Weintraub does convey the essence of most of his lemmas and theorems, and he says much to illuminate. For example, on page 101: “Everyone knows how to construct an equilateral triangle. But… gives a construction of an equilateral triangle that is different from the usual one.” And on page 130 he refers to the Theorem of Natural Irrationalities as “a result that enables us to compare Galois groups over different fields”.

However, he doesn’t say anything about how that theorem got its nickname. And on page 76, when he first defines the norm and the trace of an element in an extension, he uses “mere” equations, and I thought immediately, “Oh, the norm is the product of the conjugates, counting multiplicities. And the trace is the sum…” It wouldn’t have used up too much paper to put in something to that effect, and would give readers (or at least this reader) a better initial appreciation of those concepts. (And I wouldn’t have minded if the book went to 182 1/ 4 pages…)

On the other hand, perhaps Weintraub meant to (sometimes) give his readers space to draw these kind of conclusions for themselves , and attain that kind of appreciation. On still another hand, why withhold helpful and time-saving information? It’s a fine balance, and often a function of individual taste. Also, sometimes I felt that the statements of the lemmas, theorems, etc., could have been more pointed, and therefore clearer and “prettier”. For example, on page 94, Lemma 4.3.2 could have been stated simply, “Radical polynomials have solvable Galois groups”.

There seem to be a lot of typos and other careless mistakes. (It seems the more advanced the book, the more typos…) I hate to say this, but there are too many to list. (Of course, I might be mistaken about some of them…) The first that I found is on page 10: Four lines from the bottom, “(11)” should be “(10)”. There are indeed a large number of errors of the mis-numbering ilk. (I realize how difficult numbering properties, theorems, and so on must be as any book, in the process of being written, undergoes important changes. I’ve never written a textbook and if I ever do, it will be a huge challenge to make less than ten mis-numbering errors, as this book seems to.)

In the middle of page 22, (second line up before Lemma 2.5.2) “i.e.” should be “are”. The gravest error (although I’m sure that math-people will get past it) is, I think, on page 24. Those two lemmas as stated are, at least to me, a minor mess! I think of the first as being about extending field isomorphisms to isomorphisms between simple algebraic extensions of those fields, and the second as being about extending field isomorphisms to isomorphisms between finite algebraic extensions of those fields. To my mind, that could be mentioned, but my main concern is that, in Lemma 2.6.1, Weintraub forgot to say what f2 is. Moreover, beyond that, I feel that the Lemma could be more clearly stated, with less extraneous notation. (And in Lemma 2.6.3, very end of the second line, another typo: F should be f.)

There were errors towards the end of the book, too, and the one which threw me for a minute or two is on page 156; Lemma 5.4.1 is exactly the earlier 5.2.2, which wasn’t quite acknowledged, and I had to figure out in what sense it “generalizes” Lemma 2.6.3. (It does, but that’s not completely obvious, at least not to me.) Also, on page 157 it was my impression that he proves, in the course of a longer proof, that elements of Galois groups take roots into roots, whereas that fact has been a “given” for a very long time. I found that distracting. Finally, on page 75, the proof of Lemma A.3.2 had me worried. First he names the (one) hypothesized transposition (i, j) and then he ends the proof with “for any i, j”…? (I stopped worrying when I realized that that lemma doesn’t seem to be needed for the proof that neither the symmetric nor the alternating group on n elements is solvable, for n greater than or equal to 5, which was the thing I was most interested in. But for other readers that proof seems to need a little work.)

Here are a few slight quirks that I found refreshing: On page 93 he defines “radical polynomial”, soon explaining in parentheses, “Radical polynomial is not a standard term, but radical extension is.” And on page 118 he similarly makes occasion to say, “Abelian extension is standard terminology but symmetric extension is not.” I think this is good practice, both to use non-standard terminology and to indicate when he does (though I wish that he had followed suit on page 94, and defined “Galois group of a polynomial” as, for example, does Stewart). It was also interesting to me that his exercises appear only at the end of every chapter, and not at the end of every section. I admit that this sometimes felt like an advantage to me, since after every section I could then hurry on to the next without feeling that I should first work on the exercises. On the other hand, that probably isn’t the best way to learn and besides, once one does get to do the exercises, one doesn’t know which section to look things up in… I also think it’s curious that, instead of x as the “polynomial variable”, he uses X, and also, instead of Zn, he keeps using Z/nZ.

And I have one minor lament: I suppose it’s a sad fact of mathematical life that advanced texts have terrible indices. This one is only slightly over two pages, and contains neither “cyclotomic”, “primitive (root of unity)”, “separable”, nor even that ever-lovin’ “quintic”. After awhile it was kind of fun to test these terms, and to learn to live with this territory.

In summary, this book is a lovely exposition, doing justice to its subject. I’m still excited about having had this opportunity to revisit Galois Theory. And in the bargain, the 182 pages are not small-print.

Marion Cohen‘s poetry book about the experience of math, Crossing the Equal Sign, was reviewed on this site.

BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.



Introduction to Galois Theory.- Field Theory and Galois Theory.- Development and Applications of Galois Theory.- Extensions of the Field of Rational Numbers.- Further Topics in Field Theory.- A. Some Results from Group Theory.- B. A Lemma on Constructing Fields.- C. A Lemma from Elementary Number Theory.- References.- Index.