Self-Studying Abstract Algebra; Aye or Nay?

Question

I am a high schooler with a deep interest in mathematics, which is why I have self-studied Linear Algebra and have begun my self-study in Differential Equations. As I am a man who likes to plan ahead, I'm pondering what field of mathematics to plunge into once I've finished DE's. I am thinking of Abstract Algebra: it has always sounded mystical and intruiging to me for some reason. I have a couple of questions regarding AA:

• What exactly is Abstract Algebra? What does it study? Please use your own definition, no wikipedia definition please.

• What are its applications? Does it have a use for example in physics or chemistry, or is it as abstract as its name suggests?

• Would it be a logical step for a high schooler to self-study abstract algebra after studying LA and DE's, or is there a field of (post-high school) math 'better' or more useful to study prior to abstract algebra?

• What are some good books, pdfs, open courseware etc. on abstract algebra? links and names please.

Answer 1

Yes, abstract algebra is a logical next step for you, presuming that you have developed an appropriate level of mathematical maturity from your study of linear algebra and differential equations (e.g. if your textbooks taught theory more than computation).

It is difficult to define abstract algebra in a short MSE answer. But one essential point deserves wider emphasis: from an algebraic perspective it's crucial to forget about any internal structure possessed by the elements of the structure. Such internal structure is an artefact of the particular construction employed. Such representational information is not an essential algebraic property. It matters not whether the elements are represented by sets or not, or by sequences, matrices, functions, differential or difference operators, etc. Instead, what matters are how the elements are related to one another under the operations of the structure. Thus the isomorphism type of a ring depends only upon its additive and multiplicative structure. Rings with the same addition and multiplication tables are isomorphic, independent of whatever 'names', representations or other internal structure the elements might possess. Andy Magid emphasizes this nicely in his Monthly review of Jacobson's classic textbook Basic Algebra I. Here is an excerpt:

"This reviewer will not attempt a definition of this essence of algebraic thought. But perhaps the reader will excuse a little soft speculation. Algebra seems to be about the holistic properties of collections of things which, while they have no independent status, derive their significance from the relations and operations that exist on the collection as a whole. It makes little difference that the elements of our ring, say, are matrices, or differential operators, or formal linear combinations of group elements; in fact it can even be a positive hindrance to think of them that way. For example, consider the groups of Galois theory: it is vital, in the end, to think of these groups as groups of substitutions in the roots of the equation being examined, but the technical problems which would attend an attempt to prove the criterion for solvability by radicals while working exclusively within this particular representation would be enormous. Far better to ignore the nature of the elements and to think of the group as a thing in itself. To take another example: the notion that the cosets of a normal subgroup of a group, while they have intrinsic meaning as subsets of the original group, are best thought of as unities, as elements of a new group, the quotient group, is often the pons asinorum of the Basic Algebra course. Those who cross it successfully usually do learn to think algebraically. It is probably unfair to claim this thought mode - ignoring the essence of elements of a structure and focusing on their relations - as exclusive to Algebra. This is the basis of much modern abstraction, and not only in mathematics; see for example [Piaget: Structuralism]. But Algebra does seem to appear whenever structures dominate a piece of mathematical thought."
-- Andy Magid, Amer. Math. Monthly, Oct. 1986, p.665
-- excerpt from a review of Nathan Jacobson: Basic Algebra I

Thus abstract algebra teaches a sort of structural abstraction, which is ubiquitous in mathematics and its applications. For example, you ask if algebra has applications in physics and chemistry. One of the subjects of algebra is a general study of symmetry by way of group theory. In chemistry this applies to crystals via the study of crystallographic groups, and in physics the Lie symmetry groups of partial differential equations play fundamental roles, e.g, governing conservation laws and separation of variables. And, despite Hardy's speculations to the contrary, even very "pure" fields of algebra like number theory have found important applications (cryptography, coding theory, etc). Also algebraic geometry has many interesting applications (e.g. in robotics and control systems), especially using effective constructive techniques such as Grobner bases. These are only a few of numerous physical applications.

Answer 2

I will answer your question by steering you to a different field first. At most any college I know about, before taking abstract algebra, you would take a course called something like Math Reasoning or Introduction to Proofs or something like that, where you learn a lot of the basics of proving things. That is, such a course is a prereq for abstract algebra. So, it would probably be good to at least study some of this first, and you did not mention that at all. In linear algebra, you might learn a bit about proving things, but not all you need to know.

This class will introduce you to things like basic logic, methods of proof (including mathematical induction), set theory, perhaps $\delta$-$\epsilon$ proofs that in the U.S. we usually don't do in Calculus 1, binary relations, functions, bijections, and such. And, as you learn these basic things, you can get introduced to some fun ideas along the way.

Of course, you're talking about self studying so you're looking for a book. It looks like the book professors are using at my current institution is Mathematical Thinking by D'Angelo and West. Once you get some basics, you can read Conjecture and Proof, a book written from teaching a more advanced course on proving things at the Budapest Semesters in Mathematics. It has lots of cool topics in it.

Answer 3

Abstract Algebra generalizes the types of things you can "do algebra" on. For example, in group theory, you often work with permutations instead of numbers. Things start to look more like the algebra you're already familiar with when you get to ring theory and field theory, but there are still plenty of unusual objects in those areas.

Essentially what you do is take away as many previously held assumptions as you can and try to reduce algebraic problems to only what is actually needed. In this sense, you seek the broadest view possible, and in doing so are able to apply what you learn to a much larger range of structures.

Applications:

• Modern cryptography is entirely built off of abstract algebra, in particular from the theory of finite fields. Coding theory is also very algebraic, with both direct applications and "how to think" applications.

• Group theory is used in quantum mechanics, in fact there is arguably an entire subdiscipline of algebra devoted to this.

• Other areas of math require an appreciable understanding of abstract algebra when moving to advanced levels. Obviously anything with "algebraic" in the title is going to need it (e.g. algebraic geometry, algebraic number theory, etc.) Other examples, topology uses fundamental groups, and functional analysis uses tons of abstract linear algebra. For this reason people often say that learning abstract algebra is learning the language of higher mathematics, although some (myself included) find the field interesting for its own sake.

As for when it is right to start studying abstract algebra, you are ready as soon as you are able to write proofs. If you are comfortable with direct proofs, contrapositive/contradiction, and at least a little induction, then go for it. If not, buy this book.

If you think you're ready, my personal preference has been to study with several books at once. Many will recommend Dummit and Foote, which is a great book, but in my opinion may be too heavy for a beginner. I started on Fraleigh, which was okay. (If you are self-studying, Fraleigh may be good because there is a solutions manual available online.) Although I have not used it myself I have heard that Artin is heavily motivated by examples (very important for abstract algebra) and thus a good place to start. I don't know too much about online abstract algebra resources, but in general I think books are better. So I recommend you procure one of those, either from the library, through the back alleys of the internet, or with actual money if you have it.

Answer 4

Before foraging ahead in the standard cirriculum, I would recommend

• Learning more history of math
• Working through lots of concrete examples rather than trying to learn the theorems in their fullest generality (as they're presented in many algebra texts).

Learning history of math can provide great context and motivation for modern math, not to mention the inspirational value of the stories of great mathematical discoveries. The standard cirriculum usually offers little in the way of motivation and relatively little in the way of concrete examples, and so it's easy to spend a lot of time learning theorems and proofs without developing a good intuitive sense for what's going on. (I'll note that these my remarks here are not intended specifically for autodidacts, but rather for people who are interested in learning higher math in general).

Some algebra specific recommendations that either offer a historical perspective or a lot of simple examples are:

1. Ronald Irving's Integers, Polynomials and Rings
2. Sethuraman's Rings, Fields and Vector Spaces
3. Jean-Pierre Tignol's Galois' Theory of Algebraic Equations
4. John Stillwell's Mathematics and Its History and Elements of Algebra

I would also recommend not tying yourself down to one book in particular - it can be very beneficial to take a look at lots of different books because often of the topics in a given subject, Book A will be best for topic X while book B will be best for topic Y, etc. If you don't have access to an academic library and can't afford to buy lots of expensive math books, you can still order lots of books through interlibrary loan at your public library.

Feel free to email me if you have more questions.

Answer 5

Here is a link to Harvard Math 122 (Extension 222) taught by one of the greats - Benedict Gross.

It offers a full series of videos along with complete notes taken by a very competent GSI which makes for a very thorough presentation of the first algebra course.

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

It follows "Artin" which I think is an excellent entry point for intuitive understanding. (An echo of the above remark regarding Dummit and Foote - which is much better when you get further in rings, modules, etc.) Should you choose to go this route and buy the text, I would recommend the more recent 2nd edition.

While I'm no expert on the math curriculum, most programs entail analysis, topology, and algebra. If you haven't yet undertaken real analysis, here is a link to a free downloadable beautifully transcribed lecture course by Fields Medal winner Vaughan Jones. This too I assure you is outstanding, and may be you best next step.

https://sites.google.com/site/math104sp2011/lecture-notes

Lastly, not knowing what you used to study linear algebra, you might want to take a look at Axler's "Linear Algebra Done Right." I would do that after the real analysis and before the algebra material.

P.S. Since these materials are available at no cost, you might want to just take a taste- like chocolate, eating is believing.

Answer 6

A friend of mine showed me this free abstract algebra book (Abstract Algebra: Theory and Applications by Judson) a while back. It's fairly recent, and looks reasonably comprehensive enough. I admit, I haven't read the entire thing, but it seems like it would be a good starting point.

http://abstract.ups.edu/index.html

It contains applications as well, which I haven't seen very much in introductory abstract algebra texts.