# Preparing for the Algebra Comp

• This Comp primarily covers material from the following course:
• MATH 540

However, it also covers some material from the following courses:

• MATH 247
• MATH 347
• MATH 545

• Here are some sample Comps in this field:

• Formal syllabus. Includes topics and reference materials.

Ok, let's look at the old exams. What can we learn?

• You always choose six problems. That's the first mistake people make: If you only answer five, you're throwing away points. A partial answer on the sixth is better than nothing. If you're worried about looking foolish, you can always write a note: "I have proved that this homomorphism is injective. I don't know how to prove that it is surjective, but I believe that it is." (Of course, don't write garbage just to get points -- the graders will distinguish between irrelevant rambling and steps that contribute to a correct solution.)

• Most students feel most confident about group theory. But it will probably be impossible for you to choose six problems on group theory. Either there won't be six problems given, or the directions will force you to choose some from ring theory and linear algebra.

Some students just study group theory and hope to pass on that. This is, in theory, possible: Absolutely perfect scores on four problems on group theory would, in some years, be enough to pass the exam. But that's very, very unlikely. First, enough group theory problems are trickly enough that it will be very hard to get perfect scores. Second, 66% isn't guaranteed to pass, and even if it does, it would probably only be a C (especially since graders would look askance at a paper that only answered group theory questions).

Bottom line: You should learn some ring theory and linear algebra too.

• Linear algebra is the overlooked low-hanging fruit of the exam. Every recent exam has had a linear algebra problem on it, and the Spring 06 exam had two. Most people didn't try them. The students who did picked up big points on them. That's because they're easier than the other topics: They cover undergraduate linear algebra (MATH 247 and maybe some 347) as opposed to the other topics, which cover graduate algebra. So review your notes from MATH 247 and/or 347 and grab some easy points.

• To be safe, you should learn all the topics on the syllabus. But certain topics seem to come up on almost every exam, so these topics are definitely worthy of extra practice:
• Groups
• Counting problems in which you are given a group of order n (some fixed number) and asked to find a subgroup or element of order k. This, of course, uses Sylow Theory.
• Classification of all abelian groups of order n.
• Showing that a subgroup is normal. There may be many different strategies for this.
• Common examples: symmetric groups (Sn), alternating groups (An), dihedral groups (Dn), and cyclic groups (Zn). Even if they aren't mentioned in the questions, you may need to use them as examples in your answers.
• Rings
• Ideals and quotient rings.
• Prime and maximal ideals (or quotient rings being integral domains or fields, which is the same idea).
• Examples of the form Z (integers) adjoin the square root of a negative number, like -1 (Z[i]), -3, or -5. These examples often come up in connection with Euclidean domains, principal ideal domains, and unique factorization domains. You should know the implications and counterexamples between these ideas. Other related concepts are prime and irreducible elements.
• Linear Algebra
• Matrices.
• Vector spaces, basis, and dimension.
• Eigenvalues and eigenspaces.