Could someone here give me a rough overview of the strengths and weaknesses of Dummit and Foote's 'abstract algebra' compared with, for instance, Fraleigh's 'A first course in abstract algebra' and maybe give some advice as to which is best for my current level.
I am looking for a good book on abstract algebra (and if possible linear algebra).
Obviously as most of these texts are fairly expensive I want to know for sure which one is best for me. Could someone here give me a rough overview of the strengths and weaknesses of Dummit and Foote's 'abstract algebra' compared with, for instance, Fraleigh's 'A first course in abstract algebra' and maybe give some advice as to which is best for my current level.
I'm not yet an undergraduate, but I've read the book 'An introduction to abstract algebra' by W. Nicholson, as well as having done many of the exercises. The book seems to cover a lot of the introductory stuff for groups, rings and fields, as well as coverage of other material such as the sylow theorems and some Galois Theory. I want to move onto a book which is more advanced, though preferably one that I can successfully self study and which maybe contains the introductory stuff so I can review it (I don't $textit{own}$ my textbook, I have to give it back soon).
I am also reading some introductory analysis, but any textbook which does not reference too much analysis without explanation would be good.
If linear algebra is not contained in the book, could one also direct me to a suitable text on that, please?
Thank you
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Here are some of my suggestions.
Make sure you are familiar with the material of Nicholson's book before you reading Foote's.In my experience, it is not enough to read only once in abstract algebra.I suggest you study Fraleigh's book. You need to clarify the difference between a ring with unity and a ring without unity. Nicholson defines a ring as having a unity. This assumption creates some confusion for me when I read Hungerford's Algebra after I reading Nicholson's book.
There are many advantages in Fraleigh's book.
(a) Its exercises are in order from easy to difficult.
(b) Fraleigh teaches readers many concepts in learning algebra.For example, he says that: 'If you do not understand what the statement of a theorem means, it will probably be meaningless for you to read the proof (2/e p.xi).'Another example appears when he was teaching Lagrange's Theorem.He says: 'Never underestimate results that count something. He mentions this sentence many times throughout this book.'
(c) He compares theorems in group theory and ring theory.
(d) He emphasizes the three most important theorems in basic ring theory (p.248).
(e) He gives an excellent explanation of field extension.Especially $Bbb{Q}(x)cong Bbb{Q}(pi)$(p.270).
The advantages of Foote and Dummit's book.
(a) They give the relationship between field, E.D., P.I.D., U.F.D. and I.D. by an inculsion chain (3/e p.292).
(b) They compare the notion in module and vector space by a table (p.408).
(c) They give an excellent explanation of representation theory.(They show the similarity between $FG$-module and $F[x]$-module.
The disadvantages of Foote and Dummit's book.
(a) They usually give their assumptions in the beginning of each section.This convention often makes me wonder because when they state some theorems or exercises,they omit the assumption.
(b) They only give the algorithm of how to find the canonical rational form of a matrix.You need to refer to Goodman's Algebra and Weintraub's Algebra to understand why the algorithm works.
I recommend you read Hungerford's Algebra as an advanced text book.
(a) It has the same level as Foote and Dummit's.He clarifies many concepts which I had previously misunderstood. For example, the form of an ideal varies from ring to ring (p.123).
(b) If there is a theorem which states the $PRightarrow Q$, then he always give an example why the reversion doesn't hold.
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(c) He discuss ring without unity. I think this is important for me in the advanced ring theory. See ch. IX. The structure of rings.
In summary, if you want to be familiar with abstract algebra, you don't need to compare these books. Because in my opinion, you should read all of them (even it is still not enough).
For linear algebra, I recommend Friedberg's book.You can treat it as an easier version of Hoffman's.If you want to learn linear algebra by more geometric interpretation or intuitive aspect, then Anton's book is a good choice.
I am going to be taking a year off from my studies and would like to self study abstract algebra as it is right now the biggest gap in my math background.
I have a copy of Dummit and Foote from which I would like to study, however I realize that it contains quite a large amount of material! I would thus like to put together a list of essential topics to cover so that at the end I would have covered a similar content to a third year undergraduate course for mathematicians. One thing I would like to do if possible is get an introduction to Galois theory, it is quite mysterious to me and I would love to get acquainted to the subject.
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I know this is an ambitious project but I am quite motivated so any tips are very appreciated!
I actually did this, and so I have some experience with Dummitt and Foote that I'd like to share. First, if you're not committed to D&F, but just to getting some good basis in modern algebra, I might be inclined to recommend a different book. Let me just tell you some pros and cons about this bible of algebra.
Pros:
Tons of examples worked out. Every chapter has some general theory, followed by usually about a half dozen explicit examples. It's really good to do these examples by yourself, and then read how the book does them, or read them in the book and then try modified examples for yourself and see if you can follow the same ideas. Another useful thing to do is to try to work some examples as you read through theorems. None of this is unique to D&F but rather just good advice for reading any math book, but it goes double for D&F for reasons I'll explain below.
A billion and two exercises. This is a must for algebra, for the same reason. Practice practice practice. The problems range from routine to difficult (but none are so hard I could never work them out, if you like to be $really$ challenged you may not like this about the book). The first dozen chapters are so have solutions online (google 'project crazy project'). This doesn't include the stuff on Galois theory you're potentially interested in, but it does include everything up to the stuff on modules over PID, if I remember correctly.
It covers nearly everything one could possibly want to know. It really is an encyclopedia of algebra, at a level that's pretty accessible - it does not have the level of formalism that other books, like the one by Lang have. No knowledge of categories is required to learn from the text, and when they are introduced in the discussion of tensor products, all the relevant notions are included.
Cons:
IT'S LITERALLY HUGE. Trying to finish even a substantial portion of it (let alone all) is a near impossible task. I studied from it for a year when I was an undergrad (probably less advanced than you are now though) and I only got through like 9 chapters.
Many of the later chapters contain things that are not part of a typical first course. The two semester sequence of algebra at my university covers only up to the stuff on Galois theory and fields, which is like 13 or 14 chapters in, and skips around quite a bit (and over some topics completely), yet the book is like 20 some odd chapters long.
Dummit and Foote's style is a little deceptive. They do not do a very good job demarcating which theorems are fundamental for you to understand and which are just 'results' that come up you may want to know. It is my opinion for example, that nilpotent groups, semidirect products and other content from the later chapters on group theory are not particularly important to understanding the basic ideas of group theory, the isomorphism theorems, group actions and Sylow theory, etc.
I really like Dummit and Foote's book. But it's not the be-all and end-all book that some faculty make it out to be. There are other very good books that contain other useful perspectives and are at a variety of levels. So my advice is to use D&F as a list of topics in algebra, and a great source of exercises and examples, while hunting out other references in the areas you like. Modern algebra is an ungodly large field - it's literally one of the three main branches of math - that there is no hope to learn about everything, at least at this stage. So use it as a starting point, and as a springboard into the areas of algebra you like. Ideally you could follow a syllabus from some other place, at least until you feel like you have a sense of direction, and then just peruse many references online using particular chapters or sections of D&F as a place to get started from. A given chapter of D&F is readable (with many exercises done) in anywhere from a week to two weeks, depending on your time and maturity. This is a great way to get some exposure to some ideas, and to help you think about branching out in other directions.
Here's a short list of other texts that I like for learning about algebra that may serve as useful references.
1) Chapter 0. This book is my favorite introduction to algebra. It includes many interesting topics not present in other books, and uses category theory throughout (it has a first chapter to introduce you to it) and is by far the friendliest algebra book I've ever used. As a result it's a little slow, but you can always breeze through it just for it's perspective while working from D&F. The perspective of universal properties is really essential to how algebraists do algebra, and so is immensely useful to carry with you as you learn. It's also the only book I've ever seen which is readable by an undergraduate which actually discusses any homological algebra at all - the others in this list (and D&F, if I remember right) do discuss it, but the discussion is certainly more difficult to read. There's even a short discussion at the end about derived categories and spectral sequences, which are certainly advanced topics.
2) Grillet's 'Abstract Algebra.' Similar to Dummit and Foote, but a little less massive. If you worked through something like this, you'd be able to get to Galois theory much quicker. On the other hand, it doesn't have the wealth of examples and problems, but the exposition is clean and clear. The later chapters go in a different direction from D&F, moving towards more homological things and ending with a chapter on universal algebra. These chapters, in addition to the later chapters from D&F on representation theory could be skimmed to see what grabs your attention.
There are a wealth of other books that are well-regarded, but I can't give much feedback on them. Rotman, Jacobson, Lang, etc. Any or all of these should be used in parallel with D&F's book in order to get the most out of all of them.