Reviews (3)

A beautiful text
Fashions come and go, even in mathematics textbooks.Books from the 60s and 70s often have a set-theoretic flavor to them, because that was the fashion.Few books of any period are well written enough to resist such aging.spivak/calculus is one, and birkhoff&maclane is another.

This book strips algebra bare.I'm not talking about kiddie stuff like y=mx+c, I mean answers to the questions 'what does it mean to add two numbers together?' 'what is a real number'?Now there's no getting away from the use of sets and logic for these sorts of questions, but B&M do it with such elegance and clarity of exposition that it seems perfectly natural.And when you think about it, they're answering pretty fundamental questions; once that your school teachers glossed over.You can add 2 apples to 3 apples and count 5 apples, and maybe 2.5 and 3.5 apples make sense, but on what logical basis can you say that pi + pi is 6.28... given that you can never have exactly pi apples?Does saying you have a real number of anything make sense?

This is how algebra texts ought to be written
I have just started reading this book, and already I am
enthralled by the beauty and elegance of the authors'
exposition. Assuming nothing more than an acquaintance with
school algebra and a little geometry, they develop
the basic properties of central algebraic structures, including
rings, groups and fields. These are treated by reference to
familiar examples, such as the ring of integers and the
rational, real and complex fields. Everything that one learned
in school algebra is to be found here, though, as is to be
expected, each topic is treated at a rigorous, mathematically
sophisticated level. In the first two chapters, the properties
of the integers and rational numbers are gradually examined,
ultimately down to the definition of addition and multiplication
on the basis of Peano postulates. The authors then consider
polynomials, the real and complex numbers, vector spaces, linear
algebra and other topics.
The writing style is clear, concise and elegant, with each new
concept being carefully defined as it is introduced. The proofs
achieve a satisfying balance between detail and brevity. Indeed,
reading the proofs and completing the exercises would do much, I
am sure, to enhance a reader's mathematical facility.

If you are interested in acquiring a deeper understanding of
algebra, this book should serve as an excellent introduction.

Reviews (21)

Wedderburn, Waring and Hamilton
Not necessarily in that order.President McCosh of Princeton
waxes eloquent in his Scottish Philosophy book somewhere on
the internet, re: Dugald Stewart, Kant and Hamilton.

Hamilton is a strong vice, but clearly represented in Herstein.

Good Introduction, useful for self study
I am an engineer by training and a sales man by profession, with a a strong liking for mathematics.
I found this book to be an very readable introduction to a subject (abstract algebra), I had never been exposed to during my engineering math - other than matirx theory, which was obviously taught extensively.
The proofs are generally easy to understand, but certainly not trivial.
A pleasure to read

Reviews (1)

Isbn: 0387568018
Sales Rank: 955119
Subjects:  1. Algebra    2. Science/Mathematics   

Reviews (14)

good textbook. but....
During the lecture on the abstract algebra 1, Jacobson's book was the main textbook. Although his book is regarded as "bible" by some professors, I cannot help confessing thath i need another reference book which explains the material in more detail and more carefully. So I consult Frayleigh's algebra textbook whenever I hit upon something difficult to understand on jacobson's. Frayleigh's book is kinder, not to economize explanationsand examples. Jacobson is one of the top-class mathematian, but his fame doesn't assure his book's efficiency. His book is written too abstractively. I don't want to recommend this book to the beginers although it has the title "basic".

Excellent!
I'm truly shocked that this book has received such low reviews.This is not an easy book, but it shouldn't be faulted for that.It's clear and beautifully written, and it's been a pleasure to work through.Additionally, the chapters are divided into sections that are 'bite size' with exercises at the end of each, which has made it well suited for regular daily study.I would highly recommend it to any student with some mathematical maturity who wishes to get a good foundation in the subject.

Isbn: 6001505667
Sales Rank: 1322166

Reviews (4)

Good book but many errors
This is a good book, well-organized and contains many useful materials. However I do not recommend it as an independent study textbook (which is what I did). There are many errors, most of them are typographical but some of them are wrong numbers which are very difficult to correct, especially in the exercises given. And the exercises are very important -- the following discussions and proofs depend on some of them. Springer published a "corrected second printing" in 1999 but obviously they didn't do a good job (I purchased the new one and I got the old one in the library).

Fun but not easy
I am not sure I would take this specific book to a desert island but I would highly recommend it to anyone who is in possession of patience, self-motivation and willingness to spend time working through both the problems and any gaps that may occur within the proofs.This book can be used both by Graduate students or as a topics/independent study course for well prepared and motivated undergraduates.
In general, you can be guaranteed that the exposition present within any book by Rotman is par-excellence and this book is no exception.The material is presented in a sound and definite logical manner, which results in gently directing the reader through an 'introduction' to group theory by presenting the material in well thought out and bite sized portions.The only negative aspect of this book, which is minimal as best, is the authors choice of notation, for example the notation used to define G-sets in chapter 3 does not necessarily do the topic justice since G-sets in their own right are an important concept.But, as stated, this is the only complaint and is easily remedied.Enjoy!

Reviews (5)

Relatively readable book based on category theory
I won the Slovak translation of this book in a competition at the high school. It had many useful definitions and theorems in it and its rigorous, but readable approach seemed attractive to me and almost converted me into a rigorous mathematician, but obviously it turned out to be too abstract at the end.

However, the main problem was that several years later, because of particle-physics motivations, I tried to learn some things about the Lie groups, Lie algebras (e.g. exceptional Lie algebras), and their representations, and initially I assumed that such a big book had to contain something useful about these topics. It did not, I think.

This book is very good if you want to learn most of the rigorous math, partly based on the generalized abstract nonsense (also known as category theory), as it was known decades ago - the definitions and basic theorems about the basic algebraic structures. But you must be ready that many newest subjects - and those that are hot today - are not covered.

The book contains many problems - and these problems are solved in another book, which you may find very useful.

bold and beautiful
It has several sections not present in most introductory texts -- affine and projective geometry, multilinear algebra, and linear algebra (the latter only seen in Herstein's Topics in Algebra), category theory, and lattice theory. The first few chapters use permutations a lot for examples, later it uses matrix groups. We are talking about the 3rd edition here -- don't get an earlier edition!

THE algebra book, period.
After getting frustated by nearly all the so-called "authoritative" books on abstract algebra (Lang, Hungerford, Jacobson), I really can say that MacLane/Birkhoff is the best die-hard classic on algebra (I've already reviewed an out-of-print edition here at Amazon, but since that review is not reproduced in this edition's page, I'm doing it myself).

There's an interesting thing about the evolution of this book: the first edition has become famous among mathematicians, because it brought for the first time an elementary exposition of categories and universal constructions, directly from the horse's mouth (MacLane founded the theory of categories together with S. Eilenberg; Birkhoff was the creator of the theory of lattices), which is used as a basic tool throughout the book; it also contained unusual topics such as multilinear algebra and affine and projective spaces, but no Galois theory. The second edition has gained a chapter on Galois theory, but has lost the part on affine and projective spaces.

Isbn: 047208500X
Sales Rank: 528006
Subjects:  1. Abelian groups   

Reviews (1)

Pretty good
This book is an advanced treatment of field theory and Galois theory and is meant for those readers who have a substantial background in graduate algebra. The subject matter used to be thought of as purely mathematical, but due to the influence of the field of cryptography, it now has many applications. I only read part 1 of the book, so my review will be confined to this part.

The author begins the discussion with field extensions. One can view a field L containing another field K as a vector space over K, and the dimension of L (as a vector space) is then called the 'dimension' of L over K. If one considers a subfield K of a field M, and an additional element u in M, then there is a smallest subfield of M containing K and u. Calling this field K(u), u can be either transcendental or algebraic over K. The author then proves some elementary properties of the field K(u), showing the existence of an irreducible polynomial for u over K. This then motivates him to call a field L containing K 'algebraic' over K if every element of L is algebraic over K. Otherwise L is called 'transcendental' over K. The dimension of K(u) over K is called the degree of u over K. Finding the degree of u can be done by finding the irreducible polynomial for u. The author also proves the arithmetic relation between the dimensions of towers of fields, and this allows him to prove the famous results on the impossibility of ruler and compass constructions. For a field L that lies between fields K and M the author studies the 'stability' of L over K, meaning that every automorphism of M/K sends L into itself. The correspondence between stable fields and normal subgroups of the Galois group of M/K is proven. Splitting fields are introduced as devices to obtain fields that are normal over a given field. A criterion for a splitting field that does not involve polynomials is proven, and the author gives tools that deal with fields of non-zero characteristic, these tools motivating the definition of separability. Splitting fields are normal in characteristic 0, but one must add separability for the same to hold in characteristic p. The unsolvability of the quintic is shown via a discussion on radical extensions of fields. For a field K of characteristic 0, and for a field L lying between K and another field M, where M is a radical extension of K, the author proves in detail that the Galois group of L/K will be solvable. Then if one has a polynomial with coefficients in K, then the Galois group of this polynomial is defined to be the Galois group of a splitting field of the polynomial over K. The Galois group of the polynomial is thought of as a group of permutations of the roots of the polynomial. The author then proves that if K has characteristic 0 and L is a radical extension of K which contains a root of the polynomial, then the Galois group of the polynomial over K is solvable.

Those readers involved in cryptography will find a discussion of finite fields in Part 1. The author's goal is to find the finite fields and determine their structure. He first proves that every nth power of a prime number p will yield a field with p^n elements. The author shows that the Galois theory of finite fields is simple by proving that if K is a finite field contained in another finite field L, then L is normal over K and the Galois group of L/K is cyclic.

The author also shows how the Galois group of an equation can be found explicitly for the cubic and quartic equations. He shows first that for the Galois group of a separable irreducible cubic over a field K is either the alternating group A(3) or the symmetric group S(3). If the characteristic of K is not equal to 2, then it is A(3) if and only if the discriminant is a square in K. For a separable irreducible quartic over K, then for the degree over K of the splitting field of the resolvent cubic of this polynomial, the Galois group is S(4) if the degree is 6, A(4) if the degree is 3, V (a particular normal subgroup of S(4)) if the degree is 1, and either the group of order 8 or cyclic of order 4 if the degree is 2.

Isbn: 088385015X
Sales Rank: 249939
Subjects:  1. Algebra - General    2. Mathematics    3. Noncommunitive rings    4. Mathematics / Algebra / General   

Isbn: 0883850087
Sales Rank: 1327140
Subjects:  1. Algebra, Abstract    2. Rings (Algebra)   

Reviews (9)

the source!
This is modern Galois Theory, straight from the horse's mouth! Galois Theory is taught today using field extensions rather than by actually solving polynomials, students also learn to view a field extension as a vector space over the smaller field; both of these things were pioneered by Artin. The book also has short, clear proofs of all the main theorems. The only problem is that there are no problems to work on, so I have to say this is only a good reference for Galois Theory.

Succinct exposition of modern Galois theory by a pioneer.
Emil Artin's short book gets a mention in most texts on
Galois theory. It is very short - only 60 odd pages. Yet
it is a very clear, complete and readable account of the
essential elements of modern Galois theory. It is based
on lectures he gave over 50 years ago but you might think
it was written only yesterday and is comprehensible to
anyone familiar with current abstract algebra terminology.
And the price makes it a bargain. There are no worked
examples, exercises or index here.

Reviews (1)

Reviews (9)

WARNING:THIS BOOK IS AWFUL
Some reviews mention typos.This book has so many typos that it is certainly not useful as a tool to learn Galois Theory. I had this book for a class in Galois Theory and there is at least one mistake for every page. (some mistakes more than typos in fact)

I cannot be more sure when I say "do not purchase this book" If you do, you will regret it. Sorry I can't type more, but I have a final in my Galois Theory class that I have to study for... the book is NOT helping.

Cheers

Good, but SO many typos
Good book, with a solid foundation in Galois Theory.Two major problems: the author introduces the theory over the complex field, when he really should start out in the abtract case.Worse, the book has many, many typos.They are all over the place.Someone really dropped the ball on corrections.So while this is a good intro, you need to constantly be on the look out for errors.

Isbn: 038790171X
Sales Rank: 1042372
Subjects:  1. Algebra - General    2. Group Theory    3. Mathematics   

Isbn: 3540586628
Sales Rank: 1224427
Subjects:  1. Algebra - General    2. Mathematics    3. Science/Mathematics   

Reviews (7)

A Classic
Well, let us think about this a little bit...You want to learn Category theory, whether for some course or just for the fun of it, and now where do you turn in order to learn the necessary concepts.If you are a mathematician and have some experience, then you turn to the masters, the originators of the given subject and read their work.Sure, being the founder of a given subject does not imply that you are a good expositor and hence are capable of revealing the necessary concepts for the beginner-allow me to inform that Mac Lane is indeed as good as an expositor as he was a mathematician.For any doubters, I point you to the only other text you should read on Category theory, namely, "Category Theory" by Horst Herrlich and compare this text with Mac Lane's.Aside from that, and with respect to the text, for most beginners or interested readers I would suggest the following outline: Read 1.1-6; 2.1-3 & 8 possibly 2.4; all of 3; as for 4 skip section 3; 5.1-5; all of 8.Then, dependent upon your desires and or focus as well as your mathematical ability, it should become obvious which of the remaining topics should be read.Finally, the only other source I would recommend for learning Category theory can be found on-line using the keyword 'Awodey'.Anyways, Enjoy and good luck.

You may not need this unless you major in category theory.
I entirely agree with the reviewer Lucas Wilman.
As a book by the creator of category theory, it has extensively incorpoated relevant items.
However I don't think this is a *must read" unless you major in the subject: you will seldom need more than what is covered in a typical homological algebra course.
My inmpression is this book should be entitled "Categories for the starting/working category theorists".

Reviews (1)

Isbn: 0486638324
Sales Rank: 152896
Subjects:  1. Algebra - General    2. Lie Groups    3. Mathematics    4. Mathematics / General