Galois Theory: Lectures Delivered at the University of Notre Dame by Emil Artin

by Emil Artin, Arthur N. Milgram

Clearly presented elements of one of the most penetrating concepts in modern mathematics include discussions of fields, vector spaces, homogeneous linear equations, extension fields, polynomials, algebraic elements, as well as sections on solvable groups, permutation groups, solution of equations by radicals, and other concepts. 1966 edition.

Genres: Mathematics, Science, Nonfiction, Reference, Theory, Algebra

96 pages, Paperback

First published November 30, 1958

Reviews

[Woflmao · Rating 3 out of 5]

This is a rather old introductory text on the fundamentals of Galois theory, the theory of field extensions and solvability of polynomial equations. Nowadays, the first twenty pages can easily be skipped, as they contain a review of linear algebra that any student wishing to read this book will already have encountered in the first semester.
The main benefit of this text is that it presents the most essential results in Galois theory in very easily accessible form, with easy to follow proofs and without distractions. On the downside, a lot of relevant material is left out and will have to be found elsewhere. A unique failure of Artin's original text was the complete omission of the application of Galois theory to the solution of polynomial equations, but an appendix written by Milgram seeks to remedy this. The typesetting is abysmal, and it would be nice if the publisher could at some point find someone reset the text in a descent font.

[Jason Evans · Rating 2 out of 5]

This book pissed me off so much I wanted to fight the author. But he is dead.

[William Schram · Rating 4 out of 5]

Galois Theory by Emil Artin is a reprint of a book published in 1944. I am unfamiliar with the level of mathematics required, and I was buying Dover reprints to fill my backlog. I know so little about Galois Theory that I don't know where to begin. Thankfully, the internet exists.

Evariste Galois was a mathematician from the early 1800s. He died in a duel at the tender age of 20. The night before the fight, Galois wrote out this theory and cemented his reputation in the annals of mathematical glory.

What does the theory do? Using it allows a person to provide proof against the doubling of the cube and the trisection of an angle with only a compass and a straight edge. To be more specific, it relates fields and groups.

The positive aspects of the book are as follows; it is short. At a meager 82 pages, the book flies by. As for the negative aspects, since the book is from 1944, I don't know if they improved it since then. There might be a book out there with a better treatment of the subject, but someone would have to point me in the right direction.

I enjoyed the book. Thanks for reading my review, and see you next time.

[Jonathan · Rating 3 out of 5]

At $7.50 on Amazon now, the price is great, and the exposition is classic. But it's not that great.

First: If only someone could update the typesetting/formatting. I'm not just griping about the obvious 50's lecture-note-style straight-out-of-a-typewritter kind of typesetting. Notationally many things are suspect, such as using "a" and "α" to stand for different things in the same formula (with subscripts, used multiple times, etc. Also, unlike in TeX, "a" and "α" don't look that different). Or using "x_k" as fixed coefficients while "a_k" becomes a variable. These are nitpicks, but I believe that they prevent the exposition from being more readable.

Second: It's thin for a reason. It is not that self-contained. Perhaps back in the day knowledge of linear algebra was not that wide-spread, so Artin chooses to include it. But there are the occasional, non-trivial group theoretic results that are simply assumed (such as Cayley's theorem, which is implicitly used at least once). On the other hand, the audience of this book should have completed a first course in algebra anyway, so this is not a big problem.

Third: The book is seriously lacking in examples, and hence motivation for each lemma and theorem is not necessarily clear. It appears to me that the best way to read this book is to have a skim over the whole book to get a feel for the overall progression of the book (partly because the book lacks an index, so things may be hard to find if you don't do this). Then jump straight into Milgram's bit about "applications" (basically, Galois theory in explicit form) and read that. Only whenever things stop making sense do you go back to the main bit by Artin.

[Pat · Rating 4 out of 5]

It's always great to read directly from someone as important to our modern presentation of Galois theory as Artin. He's very articulate and concisely introduces the reader to the basic tenets of Galois theory and its original applications. The approach through basic Linear Algebra concepts makes this accessible to one who has had an introductory course in that. A more modern approach to this subject should certainly be read concurrently though.

[Aryan Prasad · Rating 1 out of 5]

The typesetting quickly gives away the age of the book. Good only as an historical artefacts. A more modern text such as Field and Galois Theory (Graduate Texts in Mathematics) (v. 167) by Patrick Morandi is much more reccomeded.