I'm wondering how Dummit and Foote (3rd ed.) would fair as a first text in Abstract Algebra. I've researched this question on this site, and found a few opinions, which conflicted. Some people said it is better as a reference text, or something to read after one has a fair deal of exposure to the main ideas of abstract algebra, while others have said it is fine for a beginner. Is a text such as Herstein's Topics in Algebra, Artin's Algebra, or Fraleigh's A First Course in Algebra a better choice?
Here's a summary of the parts of my mathematical background that I presume are relevant. I've covered most of Spivak's famed Calculus text (in particular the section on fields, constructing $\mathbf{R}$ from $\mathbf{Q}$, and showing the uniqueness of $\mathbf{R}$ which is probably the most relevant to abstract algebra) so I am totally comfortable with rigorous proofs. I also have a solid knowledge of elementary number theory; the parts that I guess are most relevant to abstract algebra are that I have done some work with modular arithmetic (up to proving fundamental results like Euler's Theorem and the Law of Quadratic Reciprocity), the multiplicative group $(\mathbf{Z}/n\mathbf{Z})^{\times}$ (e.g. which ones are cyclic), polynomial rings such as $\mathbf{Z}[x],$ and studying the division algorithm and unique factorization in $\mathbf{Z}[\sqrt{d}]$ (for $d \in \mathbf{Z}$). I have only a little bit of experience with linear algebra (about the first 30 pages or so of Halmos' Finite Dimensional Vector Spaces and a little bit of computational knowledge with matrices) though.
With this said, I don't have much exposure to actual abstract algebra. I know what a group, ring, field, and vector space are but I haven't worked much with these structures (i.e. I can give a definition, but I have little intuition and only small lists of examples). I have no doubt that Dummit and Foote is comprehensive enough for my purposes (I hope to use it mostly for the sections on group theory, ring theory, and Galois Theory), but is it a good text for building intuition and lists of examples in abstract algebra for someone who has basically none of this? Will I, more than just learning theorems and basic techniques, develop a more abstract and intuitive understanding of the fundamental structures (groups, rings, modules, etc.)? It is a very large and supposedly dense text, so will the grand "picture" of group theory, for example, be lost? I've heard it is a book for people who have some basic intuition in group and ring theory, and I hesitate to put myself in this category given my description of my relevant knowledge in the paragraph above. Do you think the text is right for me, or would I be more successful with one of the three texts I mentioned in the first paragraph?
Thanks for reading this (lengthy) question. I look forward to your advice!
Best Answer
I would absolutely recommend Artin's Algebra in your situation. Apart from the book being excellently written, one of its major advantages is that it develops linear algebra and abstract algebra in parallel. Many of the most interesting applications of groups are to geometric problems, and Artin's book is great for that.
Many other books on abstract algebra, including Dummit and Foote, have much less discussion of linear algebra. The reason for this is probably that this fits the division of university courses in the U.S. between linear algebra, taken by a wide range of students, and abstract algebra, of interest to a more restricted audience. This isn't sensible from a mathematical viewpoint.