Math 370, the first semester Abstract Algebra, is a course which emphasizes on rigorous proofs. The general goal is to study algebraic operations (such as addition, multiplication, composition, quotients, tensor product) and algebraic structures (such as groups, rings, fields, vector spaces, modules). Students are assumed to have some idea about what a proof is. Part of the goal of this course is a rigorous training in the precise mathematical language, especially what constitutes a correct mathematical proof.
Since the materials in Math 370 may look abstract and formal, examples are crucial. They offer a testing ground for the concepts and theorems. Having a large number of examples at hand helps you to develop a feeling of whether a statement is plausible or not. For instance collection of common examples of groups can be found in a previously prepared review of basic concepts in group theory. You are strongly advised to go through these examples and play with them when you learn the basic vocabulary in group theory. For instance, take a groups G in the list, and pick a non-trivial subgroup H of G, you may want to figure out the center Z(G) of G, the normalizer of H in G. Later on, for a finite group G on the list, you want to figure out the Sylow-p subgroups for each prime number p which divides the order of G. In a sense the body of mathematics is a collection of (interesting) examples. The abstract concepts and theorems are useful ways for us to organize these examples.
As you must have heard that Math 370 is more demanding than all the calculus courses, in many aspects. You cannot expect to be able to do a homework problem by looking up a similar example in the textbook. We will say more about "the textbook" later. The most important part of your learning process for this course is doing the problems. It is often said that "you have not learned anything if you cannot do the problems".
The majority of homework problems are about examples of the the abstract concepts in action, so that in principle you should be able to make substantial progress if you understand the meaning of the key words. Let us illustrate this point with the following problem: find all homomorphisms from a cyclic group of order 3 to a cyclic group of order 6. How can one approach this problem? Well, first you tell yourself that this problem can be determined in a finite amount of time: Every such homomorphism can be regarded as a function, from a set with 3 elements to a set with 6 elements, and there are 216 such functions. This is a start, and if you are persistent enough you can go through the list of 216 possibilities, and decide which ones are homomorphisms from the definition. So already you know that in principle you can do it. Then you remember that a homomorphism sends the unit element of the source group to the unit element of the unit element of the target group. This cuts down the number of possibilities to 36; it is not too bad to sort out among the 36 candidates which ones are homomorphism. If you use other defining properties of homomorphisms, you can further cut down the number of possibilities, In the lab sessions you will be able to compare other people's solutions with your own. If you see a solution more elegant than your own, analyze it. You have the option of talking to other students in the class, but you should always try the problems yourself first! Your first approach may be awkward in retrospect, and that is all right. But NEVER just give up!The presentation in the lectures will be different from that of the textbook. You are encouraged to ask question and take notes. Then work out your notes after class and compare with the approach in the textbook. You are sure you know the material only when you can explain the proof to yourself and your friends without looking at your notes or books.
Many people like to have a "textbook" to study from.
The "official textbook" is Artin's Algebra;
We use Artin mostly as a reference;
the lectures will be logically self-contained.
If you take good notes you can learn the materials without any textbook,
at least in theory.
Artin's book contains more material than can be covered in 370 and 371.
Artin's book is not as formal as in some other textbooks such as
Dummit/Foote or Herstein.
At the beginning you might have to read a theorem several times before you
understand the statement and the proof. The rewards will justify the efforts.
However if you find Artin not to your liking, here are some alternatives:
Dummit and Foote has also been used in the past for Math 370 and
Math 371; it is very carefully written and contains a large number
of exercises.
Fraleigh has also been used before; it does not go as far as Artin or
Dummit/Foote.
The other two books have also been used before, sometimes
in conjunction with Artin.
Below are some other well-known general algebra texts: