How does Dummit and Foote's abstract algebra text compare to others?

Question

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

Answer 1

Here are some of my suggestions.

1. 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.

2. 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).

3. 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.

4. 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.

5. 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 $P\Rightarrow Q$, then he always give an example why the reversion doesn't hold.

   (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.

6. 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).

7. 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.