This course is an introduction to modern algebra, including basic number theory, rings, fields and groups. Sometimes there is a second semester course covering topics such as representations of finite groups, modules over commutative rings, and/or Galois theory.
Prerequisites: Linear algebra, and a readiness to read and write proofs.
Time: MWF 10 AM
Lecturer: Rebecca Lehman
Office: Gibson 412. Office Hours T-Th 10 AM-noon; other times by appointment
Required text: Shifrin, Abstract Algebra: A Geometric Approach, Prentice Hall 1996.
Format: Mostly lecture, but may include some group problem solving or seminar-style presentations depending on class size and student interest.
Problem Sets: 20%
Midterm: 20%
Final: 40%
Participation/possible final project/Other: 20%
This is not a freshman class, and problem sets are hard. I will use my judgement as the semester goes on, but I can tell you right away that an A will not require a 90%. If I see you working hard, you will pass. If I see you working hard and learning math, you will get a good grade.
Initial Problem Set: Please write up these problems by Wednesday September 5. They are not for a grade, but I will read your answers and give you feedback. The point is to give you a taste of writing proofs and to give me a sense of where you are.
Appendix A, problems A.1.4, A.2.1, A.2.9, A.3.6, A.3.9.
Don't Panic! Math problems are supposed to be hard! You should not expect to be able to solve them right away. Look at them for an hour or so, then set them aside for a few days and try again.
Mathematicians talk to each other, so I expect you to talk to each other and to me. When you do write up the problem sets, I will expect you to write them in your own words and give credit where credit is due, but collaboration is considered a good thing, not a bad thing.
The following problems are mostly for your own edification and enjoyment, but you will not learn much in this course unless you at least look at most of them. I will collect three problem sets during the semester, consisting of a sampling of problems (1-2 per week). These will be for a grade and will be marked with *. I will announce at least a week ahead of time when problem sets are due, but you will do better if you try to keep up with them week by week.
Problem Set 1: Due October 8
Lecture 1: 1.1.5*, 1.1.12, 1.1.14.
Lecture 2: 1.2.1, 1.2.12, 1.2.16.
Show that there do not exist x and y such that x+y=100 but gcd(x,y)=3. Show that if n is composite, then it has a prime factor less than or equal to the square root of n.
Lecture 3: 1.3.4, 1.3.14, 1.3.25. (Optional/for fun: 1.3.37-1.3.38)
Lecture 4: 1.3.21*, 1.3.22, 1.3.35, 1.3.36
Lecture 5: 1.4.3, 1.4.8*, 1.4.9, 1.4.11, 1.4.18.
Lecture 6: 2.1.6, 2.1.8, 2.1.13.
Monday: No lecture. Problem session: 1.2.12, 1.3.14, 1.4.9, 1.4.11.
Lecture 7: 2.2.6, 2.2.7*, 2.2.13, 2.2.14, 2.2.17.
Lecture 8: 2.3.4, 2.3.7, 2.3.17*, 2.3.20.
Give another proof of Corollary 3.4 using multiplication in polar coordinates. Use Euler's formula to show that the exponentials of all rational imaginary numbers are roots of unity. Show that the exponentials of irrational imaginary numbers have absolute value 1 but are not roots of unity.
Lecture 9: 2.4.8, 2.4.11.
Lecture 10: 2.5.1, 2.5.2, 2.5.6, 2.5.10*, 2.5.16, 2.5.17.
Problem Set 2: Due October 26
Lectures 11-12: Prove that F(x) is a field if F is a field. Do problems 3.1.1, 3.1.4, 3.1.6*, 3.1.9, 3.1.14, 3.1.20*.
Lecture 13: 3.2.1, 3.2.5, 3.2.7*, 3.2.13, 3.2.16. Prove explicitly that if R is a subring of K, and a is a root in K of a polynomial in R[x], then R[a] is a ring.
Lectures 14-15: 3.3.3, 3.3.5*, 3.3.8, 3.3.9, 3.3.10, 4.1.1.
Wednesday: Problem Session/Exam Review: These problems are intended to be similar to those you will find on Friday's exam: 1.2.1, 1.3.7, 1.4.2, 1.4.4, 2.2.4, 2.2.5, 2.3.8, 3.1.1, 3.1.10, 3.2.1, 3.3.3. Do NOT bother doing all parts of all the long problems. Do parts a-c or so, or pick one at random, and make sure you have an idea how to do the other examples. Come to class prepared to either explain the solution to each problem or ask an intelligent question about it. If you can do these without looking anything up, you will ace the exam.
Lecture 16: 4.1.2, 4.1.6, 4.1.7, 4.1.11, 4.1.16* 4.1.17, 4.1.19. Ring homomorphisms are extremely important. Please try to do all the exercises here.
Lecture 17: 4.1.18, 4.1.21, 4.2.3, 4.2.7, 4.2.13*, 4.2.17, 4.2.19. These are also very important.
Lecture 18: 4.3.3, 4.3.5, 4.3.8, 4.3.12, 4.3.18.
Problem Set 3: Due November 26
Lecture 19: 6.1.4. List all subgroups of the quaternion group and their orders, and all subgroups of the triangle group S_3 and their orders. 6.1.5*, 6.1.7, 6.1.11, 6.1.14, 6.1.15, 6.1.16, 6.1.22.
Lecture 20: *Prove that a group homomorphism is an isomorphism if and only if it has an inverse homomorphism. 6.2.5, 6.2.10, 6.2.15, 6.3.8, 6.3.9.
Lecture 21: 6.3.2, 6.3.6, 6.3.7, 6.3.10, 6.3.16, 6.3.23, 6.3.29.
Lecture 22: Here are some exercises about product groups and permutations: Products & Permutations, PDF. Products & Permutations, LaTex Source. Here are solutions: Products & Permutations Solutions, PDF. Products & Permutations Solutions, LaTex Source.
Lecture 23: 6.4.4, 6.4.10, 6.4.17*, 6.4.19.
Lecture 24: 6.4.8, 6.4.9, 6.4.15*, 6.4.16, 7.1.1, 7.1.3, 7.1.7.
Lecture 25: 7.1.4, 7.1.9, 7.1.14*, 7.1.18, 7.2.4, 7.2.5.
Lecture 26: 7.2.7, 7.2.8*, 7.2.9, 7.2.10, 7.2.12, 7.3.3, 7.3.4, 7.3.6, 7.3.9, 7.3.10, 7.3.12, 7.3.14.
Friday: Problem Session: 6.4.9, 7.1.4, 7.2.9, 7.2.12, 7.3.4, 7.3.14.
Lectures 27-28: 7.4.7, 7.4.10*, 7.4.12, 7.4.13, 7.4.23.
Final Review Problems: I will not collect any more "required" problem sets, to leave you free to work on your in-class presentations and review what you need to for the final.
The last day of classes is Friday December 7. If you want to show me any of the problems here (or any of the "optional" problems from the entire class) by Wednesday December 5, I will be happy to correct them and discuss them with you.
Lecture 29: 5.1.2, 5.1.4, 5.1.11, 5.1.12, 5.1.18, 5.1.19.
Lecture 30: 5.1.20-25. Prove your favorite linear algebra theorems over an abstract field. (A detailed discussion of abstract linear algebra can be found in Artin's Algebra, Chapters 3-4).
Lecture 31: 5.3--all the problems are good. You should be able to do 1-5 without difficulty; 6-8 are intended to be more of a challenge.
Final Exam Reviews: The exam will be Monday December 10 at 1:30 pm. On Wednesday and Friday we will discuss problems and review for the final exam. Explicit topics will be taken from chapters 4-7, although we will make use of the earlier material, especially polynomial rings and the integers mod p. Here is a review sheet: Final Exam Review.
Suggested Review Problems: You can ask me any questions you want from the entire semester. If you don't have questions, we'll go over these: 4.1.2, 4.1.17, 4.2.3, 5.1.18, 5.3.4, 5.3.5, 5.3.6, 6.2.10, 6.3.7, 6.4.8, 6.4.9, 7.1.9, 7.3.6, 7.3.10.
Exam Solutions: Here are the solutions to the final exam: Final Exam Solutions
Here is an old solution set of mine for use as a template: Sample Problem Set. If you're just starting out, ignore all the headers above "\begin{document}," then insert your own problems and solutions into the template.
This "Simplified Introduction" is a helpful reference guide: Simplified Introduction to LaTeX.
More LaTeX resources are here: MIT Math Resources.