Math 113, Section 005
Lecture: Tuesday and Thursday from 8-9:30 in 310 Hearst Memorial Mining Building
Office Hours: Monday 2:30-4 and Tuesday 1-2:30, in my office, 735 Evans. During the week of December 4, I will have office hours on Tuesday and Thursday from 1-3. During the week of December 11, I will have office hours on Tuesday from 1-3 and on Wednesday from 3-4. I am also available by appointment during both of these weeks -- just send me an email.
GSI: Justin Chen will hold office hours Tuesdays 11-3, Wednesdays 2-7, and Fridays 11-12 in 959 Evans.
Textbook: A First Course in Abstract Algebra by John B. Fraleigh, 7th edition, Addison-Wellesley
Information for students:
- The syllabus. Please read it carefully.
- DSP students should come see me as soon as possible, even if you do not have a letter yet.
- Guidelines on what to do if you think you have an extracurricular that interfere with this course. In particular, you must speak with me no later than the end of the second week of class.
- An introduction to writing proofs, courtesy of M. Hutchings. I strongly suggest you read this through before completing your first homework assignment.
- How to get an A in this class, courtesy of K. Mann.
- See below for Study Tips (for any upper level math class), courtesy of M. Hutchings.
Course Goals: In previous courses you have seen many kinds of algebra, from the algebra of real and complex numbers, to polynomials, functions, vectors, and matrices. Abstract algebra (most mathematicians would just call this "algebra") encompasses all of this and much more. Roughly speaking, abstract algebra studies the structure of sets with operations on them. We will study three basic kinds of "sets with operations on them", called groups, rings, and fields.
A group is, roughly, a set with one "binary operation" on it satisfying certain axioms which we will learn about. Examples of groups include the integers with the operation of addition, the nonzero real numbers with the operation of multiplication, and the invertible n by n matrices with the operation of matrix multiplication. But groups arise in many other diverse ways. For example, the symmetries of an object in space naturally comprise a group. After studying many examples of groups, we will develop some general theory which concerns the basic principles underlying all groups.
A ring is, roughly, a set with two binary operations on it satisfying certain properties which we will learn about. An example is the integers with the operations of addition and multiplication. Another example is the ring of polynomials. A field is a ring with certain additional nice properties, such as the rational and real numbers. At the end of the course we will have built up enough machinery to prove that one cannot trisect a sixty degree angle using a ruler and compass.
In addition to the specific topics we will study, which lie at the foundations of much of higher mathematics, an important goal of the course is to expand facility with mathematical reasoning and proofs in general, as a transition to more advanced mathematics courses, and for logical thinking outside of mathematics as well.
Homework: Homework will be assigned weekly. These will be posted on the course web page no later than midnight on Thursday and will be due at the beginning of lecture on the following Thursday. Two to three problems will be graded on each assignment.
Late homework will not be accepted under any circumstances. If you are late to class on Thursday, your homework will be late and will not be graded. In the case of extended illness, you must contact me as soon as possible. If you know you will miss a Thursday lecture, you must arrange to turn your homework in ahead of time.
You are encouraged to discuss the homework assignments with your classmates, but you must write up the solutions entirely on your own. That is, your assignment should be your own work, written in you own words (i.e., by yourself without consulting someone else's solution). Plagiarism and copying (from other students, the internet, etc.) are not tolerated under any circumstances.
- All solutions should be written in complete, grammatically correct English (or at least a very close approximation of this) with mathematical symbols and equations interspersed as appropriate. These solutions should carefully explain the logic of your approach.
- All proofs must be complete and detailed for full marks. Avoid the use of phrases such as `it is easy to see' or `the rest is straightforward', you will likely be docked marks. Proofs in your homework should be clear and explicit and should be more detailed than textbook proofs.
- If the grader is unable to make out your writing then this may hurt your mark. See the course website for additional tips on writing mathematics.
Study Tips (for any upper level math class):
- It is essential to thoroughly learn the definitions of the concepts we will be studying. You don't have to memorize the exact wording given in class or in the book, but you do need to remember all the little clauses and conditions. If you don't know exactly what a UFD is, then you have no hope of proving that something is or is not a UFD. In addition, learning a definition means not just being able to recite the definition from memory, but also having an intuitive idea of what the definition means, knowing some examples and non-examples, and having some practical skill in working with the definition in mathematical arguments.
- In the same way it is necessary to learn the statements of the theorems that we will be proving.
- It is not necessary to memorize the proofs of theorems. However the more proofs you understand, the better your command of the material will be. When you study a proof, a useful aid to memory and understanding is to try to summarize the key ideas of the proof in a sentence or two. If you can't do this, then you probably don't yet really understand the proof. (When I was a student I had a deck of index cards; on each card I wrote the statement of a theorem on one side and a summary of the proof on the other side. Very useful!)
- The material in this course is cumulative and gets somewhat harder as it goes along, so it is essential that you do not fall behind.
- If you want to really understand the material, the key is to ask your own questions. Can I find a good example of this? Is that hypothesis in that theorem really necessary? What happens if I drop it? Can I find a different proof using this other strategy? Does that other theorem have a generalization to the noncommutative case? Does this property imply that property, and if not, can I find a counterexample? Why is that condition in that definition there? What if I change it this way? This reminds me of something I saw in linear algebra; is there a direct connection?
- If you get stuck on any of the above, you are welcome to come to my office hours. I am happy to discuss the material with you. Usually, the more thought you have put in beforehand, the more productive the discussion is likely to be.
Course Calendar: I will post all reading and homework assignments here, as well as any handouts passed out in class. I will also post solutions to the homework assignments here, shortly after the assignment has been turned in. All reading and homework assignments should be completed before class on the day they are posted. For example, when you see "Thursday, 8/31: read Sections 0 and 1, Homework#1 due", this means that you should have carefully read Sections 0 and 1 of the textbook and completed Homework #1 before class on Thursday 8/31.
- Tuesday 8/29: read Section 0, 2, 4 (only through Example 4.14)
- Thursday 8/31: Finish reading Section 4. Homework #1 is due at the beginning of class. Here is a discussion of how to fix the bijective map from R^2 to R^1 that we discussed in class. Solutions to Homework #1.
- Tuesday 9/5: Read Section 5.
- Thursday 9/7: Read Section 6. Homework #2 is due at the beginning of class. Solutions to Homework #2.
- Tuesday 9/12: Read the following supplementary notes on finite cyclic groups.
- Thursday 9/14: Read Section 8. Homework #3 is due at the beginning of class. Solutions to Homework #3.
- Tuesday 9/19: Read Section 9. Quiz #1 will be given during the first 30 minutes of class. Quiz #1 will cover all material up to and including Section 6 (Cyclic groups). Solutions to Quiz#1.
- Thursday 9/21: No reading. Homework #4 is due at the beginning of class. Solutions to Homework #4 (these solutions have been corrected; the original file had the solution to Exercises 8#17, which was not assigned, instead of Exercises 8#18, which was.)
- Tuesday, 9/26: Read Section 11.
- Thursday, 9/28: Read Section 10. Review the definitions of an equivalence relation and a partition from Section 0. Homework #5 is due at the beginning of class. Solutions to Homework #5.
- Tuesday, 10/3: Read Section 14.
- Thursday, 10/5: No reading. Homework #6 is due at the beginning of class. Solutions to Homework #6.
- Tuesday, 10/10: Read Section 15. Midterm Review Sheet. Solutions to the midterm review. I suggest you try this without looking at your notes or the textbook first. Homework #7a does NOT need to be turned in. These are just good additional practice problems for you on quotient groups. Solutions coming soon!
- Thursday 10/12: Review for midterm; come to class with any questions you have. Homework #7 is due at the beginning of class. Solutions to Homework #7.
- Tuesday 10/17: Midterm #1 will be given. Solutions to the midterm. Covers all sections of the book through Section 15 (except for simple groups), except for Sections 7, 12 (which we skipped), as well as normalizers, centralizers, and the center of a group (see your notes), and everything we did about homomorphisms in class (we did this slightly differently than the book). Everything in those sections of the book, everything we discussed in class, and everything you did in your homework is fair game for the exam.
- Thursday, 10/19: No homework due today.
- Tuesday, 10/24: Read Section 18.
- Thursday, 10/26: No reading. Homework #8 is due at the beginning of class. Solutions to Homework #8.
- Tuesday, 10/31: Read Section 19.
- Thursday, 11/2: Read Sections 20 and 21 (note: we will not be covering Section 20 in class, but you are still responsible for the material -- read it carefully!). Homework #9 is due at the beginning of class. Solutions to Homework #9.
- Tuesday, 11/7: Read Sections 22 and 23. Here is a (relatively) elementary proof Wedderburn's Theorem -- take a look if you're interested!
- Thursday, 11/9: No reading. Homework #10 is due at the beginning of class (file updated 11/3). Solutions to Homework #10.
- Tuesday, 11/14: Read Section 26.
- Thursday, 11/16: Read Section 27. Homework #11 is due at the beginning of class. Solutions to Homework #11. Quiz #2 will be given during the first 30 minutes of class. The quiz will cover Sections 18-23. Note that we did not cover Section 20 in class, but you are responsible for the material; you will likely need some of the facts in the chapter for the quiz, mostly as they relate to polynomials. Solutions to Quiz #2.
- Tuesday, 11/21: Read Section 29.
- Thursday, 11/23: NO CLASS.
- Tuesday, 11/28: Read Sections 30 (this is a quick review vector spaces -- we will not be covering this material in class, but you are responsible for it) and 31.
- Thursday, 11/30: No reading. Homework #12 is due at the beginning of class. Solutions to Homework #12.
- Thursday, 12/14: Final Exam from 7-10 pm, in 310 HMMB (our usual classroom). The final exam will cover Sections 1-11, 13-15, 18-23, 26, 27, 29-31 of Fraleigh, as well the extra material on normalizers, centralizes, the center of a group, and group homomorphisms (which we did slightly differently than in the textbook). Everything from those sections in the book, everything from the homework, and everything we discussed in class is fair game for the final. Final exam review sheets: "Give an example of" problems, and Proofs. Part 1 and Part 2 of solutions to the Proofs. Corrected solution to #3 on the proofs (with additional details on finding the irreducible polynomial). Solutions to the "Give an example of" problems.