Math 113 final exam review

The final exam will be on the morning of Saturday, May 24th. The final will be similar to the midterms, but longer. Hopefully its difficulty will be somewhere in between that of the first and second midterms; however it is hard for me to judge this in advance. Although later parts of the course will be emphasized more, the final will cover the entire course.

Here are some suggestions for studying.

Make sure you understand how to do the homework and midterm problems, as similar problems may appear on the final. Study the posted solutions and compare them with what you did. Note that the posted solution is usually not the only correct solution. Also, not every homework problem was carefully graded due to time constraints, so just because you didn't lose points on something doesn't mean that it is right.
If you want more practice you could try doing additional problems from the book in the chapters that we covered. For review of the material in the last couple of lectures, the following problems might be helpful: Chapter 32, problems 2 and 3; Chapter 33, problems 8, 10, 11, 14.
The following are some of the kinds of questions that might be asked on the final.
- Decide whether two given (groups, rings, fields, vector spaces) are isomorphic. For example, are the fields Q(sqrt(2)) and Q(sqrt(3)) isomorphic? Hint: an isomorphism between these fields, if it exist, must fix Q.
- How many homomorphisms are there from one given (group, ring, field) to another?
- Understand a quotient group or quotient ring. (Note that there is no such thing as a quotient field. Why not?)
- Analyze the characteristics of some given object. For example, is some group abelian or cyclic? Is some ring an integral domain or a field? What is the order of a given element of a given group? Is some polynomial over some field irreducible? What is the degree of some field extension?
- Prove that something is well-defined. (For example, see the proof that addition and multiplication on a quotient ring are well defined.)
- Prove some simple general statement that can be deduced in a couple of sentences from the definitions or from known theorems.
It is not necessary to memorize every proof that was presented in class, although it wouldn't hurt to learn a few proofs that seem most important and interesting. It is definitely important to know the definitions and the statements of the results, and to be able to work with these in examples and simple proofs. One can think of the proofs in class and the book as good examples of how to reason with the ideas of this course.
If you get stuck on any of the above, please feel free to come to my office hours the week before the final. The times for these office hours will be announced at the beginning of that week.