**Math 113, Section 005**

**Lecture:**Tuesday and Thursday from 8-9:30 in 241 Cory Hall. During RRR week, I will hold review sessions during the usual class time.

**Office Hours:**Tuesdays 9:45-11:45 and Wednesday 11-12, in my office, 735 Evans.

During RRR week, my office hours are as follows:

- Tuesday, 5/1, 11-2
- Thursday, 5/3, 9:30-12
- Wednesday, 5/9, 10-1
- Thursday, 5/10, 3-5

**GSI:**The GSI for all sections of Math 113 is Tao Su. He will hold office hours every week Monday-Friday from 5-7 pm in 940 Evans. If these hours change, the change will be posted at math.berkeley.edu/~taosu/113_2018S.

**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.

**Exam and quiz schedule:**

- Quiz #1: Tuesday, February 13, during the first 30 minutes of class.
- Midterm Exam: Tuesday, March 13
- Quiz #2: Thursday, April 19, during the first 30 minutes of class. The quiz covers all material in Sections 18, 19, 22, and 23.
- Final Exam: Thursday, May 10, from 7-10 pm (according to the Final Exam Group Guidlines)

**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.

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 this web page (below) 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 credit. Avoid the use of phrases such as `it is easy to see' or `the rest is straightforward', as you will likely lose points. 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.

**Reading Schedule:**Outside reading is an essential part of this course. Reading assignments will either be sections from the textbook or a pdf with material from other sources. All reading assignments will be posted here, and they should be completed

*before class*on the day they are posted.

- Thursday, 1/18: Sections 0, 2, and Equivalence relations and partitions.
- Tuesday, 1/23: Section 4 (through Example 4.14).
- Thursday, 1/25: Finish Section 4, and read the section on "Integers mod n" in this file (which is pages 20-22). Note that the file contains an entire section of a book, but you only need to read the last 3 pages. If you are unfamiliar with any of the notation used, it is defined earlier in the section, which is why I included the entire file.
- Tuesday, 1/30: Section 5.
- Thursday, 2/1: Cyclic groups (you can ignore examples 1.3.7 and 1.3.8, since they involve groups we have not yet defined)
- Tuesday, 2/6: Homomorphisms. Optional additional reading: Multiple people have asked where they can read more about dihedral groups, since our book does not discuss them in detail. Here is a pretty good discussion of dihedral groups. The notation in this reading is slightly different than what we used in class -- the group is called D_{2n} instead of D_n, the elements we call "rho" and "tau" (the generators) are called "r" and "s. Also, this reading discusses group presentations toward the end, which we probably won't cover in our class, so you can ignore that part, if you wish.
- Thursday, 2/8: Sections 8 and 9.
- Tuesday, 2/13: No additional reading; review sections 8 and 9. Optional reading on roots of unity (start on page 76)
- Thursday, 2/15: Section 11, up to but not including "The Structure of Finitely Generated Abelian Groups."
- Tuesday, 2/20: Finish section 11.
- Thursday, 2/22: Section 10.
- Tuesday, 2/27: Section 14 up to but not including "The Fundamental Homomorphism Theorem."
- Thursday, 3/1: Finish Section 14.
- Tuesday, 3/6: Section 15.
- Thursday, 3/8, and Tuesday, 3/13: No reading.
- Thursday, 3/15: Section 18.
- Tuesday, 3/20: Review the end of section 15, starting with "Simple groups."
- Thursday, 3/22: Section 19. Optional reading describing how to use the fundamental theorem of finitely generated abelian groups to calculate quotient groups of infinite abelian groups (written by Prof. Williams).
- Tuesday, 4/3: Sections 22 and 23, up to but not including "Irreducible polynomials." Optional reading which gives a proof of Wedderburn's theorem. A second optional reading which gives a proof of the fundamental theorem of finitely generated abelian groups; it starts on p.25.
- Thursday, 4/5: Finish section 23, and Section 26, up to but not including "Factor (Quotient) Rings."
- Tuesday, 4/10: Finish Section 26.
- Thursday, 4/12: Section 27.
- Tuesday, 4/17: Review Section 27.
- Thursday, 4/19: Read Section 29 up to but not including "Simple Extensions."
- Tuesday, 4/24: Finish Section 29 and read Section 30 (this is a review of vector spaces; we will not cover this material in class, as you should have already seen it in 54 or 110).
- Thursday, 4/26: Section 31.

**Homework Assignments:**

- Thursday, 1/25: Homework #1 due. Solutions to Homework #1.
- Thursday, 2/1: Homework #2 due. Solutions to Homework #2.
- Thursday, 2/8: Homework #3 due. Solutions to Homework #3.
- Thursday, 2/15: Homework #4 due. Solutions to Homework #4. Solutions to Quiz #1.
- Thursday, 2/22: Homework #5 due. Solutions to Homework #5.
- Thursday, 3/1: Homework #6 due. Solutions to Homework #6.
- Thursday, 3/8: Homework #7 due. Solutions to Homework #7.
- Thursday, 3/15: No homework due this week, due to the midterm.
- Thursday, 3/22: Homework #8 due. Solutions to Homework #8. See the "Midterm Exam Information" section for solutions to the midterm.
- Thursday, 4/5: Homework #9 due. Solutions to Homework #9: Part 1 and Part 2.
- Thursday, 4/12: Homework #10 due. Solutions to Homework #10.
- Thursday, 4/19: Homework #11 due. Solutions to Homework #11.
- Thursday, 4/26: Homework #12 due. Solutions to Homework #12. Solutions to Quiz #2.

**Midterm Exam Information:**The midterm will be held on Tuesday, 3/13, and will take all of class time. The topics include everything we have done in class through Tuesday, 3/6. See the reading schedule above for the list of sections of the book and outside reading you should be familiar with (note that only the first half of section 15 is on the exam; everything up to but not including the subsection titled "Simple groups"). Remember that many important theorems/facts are done in the homework rather than in the reading, so be sure you know what you have proved on the homework. As part of your studying, I suggest you make three lists: one of groups we have discussed (as well as facts about those groups), one of definitions (make sure you know these word for word!), and one of theorems/facts (these should come from the reading, homework, and your class notes). You may use any theorem/fact that from the reading, homework, or class without proof on the exam; either quote the theorem by name, if it has one, or simply say "We showed that..."

The exam will include T/F problems, "give an example of" problems, calculations (for example, the order of an element in a group or quotient group, find the inverse of an element, find the order of a group or quotient group), as well as two short proofs and one long proof. There will be a few other short answer questions as well, such as: draw a subgroup diagram, interpret a group table, use the fundamental theorem of finitely generated abelian groups to identify the isomorphism type of a quotient group, determine if a subgroup of a group is normal, or determine if there are any non-trivial homomorphisms between two groups (this list is not meant to be exhaustive; there may be other types of short answer questions, as well!). It is likely that I will include at least one proof (long or short) that you have seen before, either on the homework, in class, or in the reading.

- Review sheet for midterm. Solutions to the review sheet.
- Here are the extra problem we did in class on Thursday (along with the problems I prepared but we did not have time to do). I did not write down the order that we did the problems in class, so the numbering may be slightly different. Here are the solutions.
- Here are the solutions to the Midterm Exam.

**Final Exam Information:**The final exam will be held on Thursday, 5/10, from 7-10 pm in 241 Cory (our usual classroom). The topics include everything we have done during the semester, in the reading and in class. See the reading schedule above for a list of the material you are responsible for. Note that there are some topics in the reading that we did not discuss in class. Such topics are fair game for the midterm with the following exceptions: (1) the material in Section 31 after the heading "Algebraically Closed Fields and Algebraic Closures" will not be on the exam; (2) the material in Section 30 (Vector Spaces) will only be tested in the context of Section 31, that is, as it pertains to field extensions.

Remember that many important theorems/facts are done in the homework rather than in the reading, so be sure you know what you have proved on the homework. As part of your studying, I suggest you make three lists: one of groups we have discussed (as well as facts about those groups), one of definitions (make sure you know these word for word!), and one of theorems/facts (these should come from the reading, homework, and your class notes). You may use any theorem/fact that from the reading, homework, or class without proof on the exam; either quote the theorem by name, if it has one, or simply say "We showed that..."

The exam will include T/F problems, "give an example of" problems, calculations (for example, the order of an element in a group or quotient group, find the inverse of an element, find the order of a group or quotient group, list the units in a given ring, find all prime and maximal ideals of a given ring, find the order of a field extension, etc.), as well as several short proofs and 2-3 long proofs. There will also be short answer questions such as: draw a subgroup diagram, determine if a map is a (ring or group) homomorphism, list all ideals of a ring, describe the elements in an extension field, etc. This list is not meant to be exhaustive; there may be other short answer questions, as well. I will include at least one long proof from the review sheet, and it is likely that I will include at least one problem that you have seen before in the homework or in class. Be sure to review your homework assignments and check your answers with the solutions posted above.

- Review Sheet #1: "Give an example of" problems. Solutions to Review Sheet #1.
- Review Sheet #2: Proofs. Solutions Part 1, Part 2 (#16 corrected on 5/9), and Part 3 to Review Sheet #2.