Algebra 1: Spring 2012

General

This is the webpage for the course Algebra 1 in Spring 2012 (MATH-UA.0343-1) taught by Assaf Naor. The teaching assistant is Lukas Koehler. Both our emails are lastname at cims dot nyu dot edu.

Assaf Naor's office is WWH 1113, and his phone number is (212) 998-3382.

Lukas Koehler's office is WWH 510, and his phone number will be determined later.

Time and place: The regular meeting times are Mondays and Wednesdays 3:30-4:45 in Silver 706. The recitations are Mondays 8:00-9:15 in WWH 202.

Office hours: Assaf Naor's office hours are Wednesdays 11:00-12:00 in 1113 WWH. Lukas Koehler's office hours are Mondays 5:00-6:00 in 605 WWH. Both of us are also available to meet by appointment.

Textbook: Herstein's book, Topics in Algebra.

Supplementary notes: We will also occasionally use the following supplementary notes of F. P. Greenleaf. 1) Integers , 2) Groups , 3) Transformation groups , 4) Permutation groups , 5) Structure of groups , 6) Universal objects .

Syllabus: Topics in Groups, Rings and Fields, chapters 2,3,5 in Herstein. According to time we may cover further advanced topics. I will post here a short description of each lecture and the relevant parts of the book as the course progresses.

Homework: Homework will be assigned weekly, and posted here by the Thursday of each week. Solutions must be handed in by the following Wednesday at noon, in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH), or by emailing the solutions to Lukas Koehler. Corrected assigments will be returned during recitations or in Lukas Koehler's office hours. For fairness reasons, late homework will not be accepted, but the lowest homework grade will be dropped to compute the final grade.

Grading: The final grade will be based on homework (20%), two quizzes (20%), a midterm exam (25%) and a final exam (35%).

You are welcome to collaborate with other students on the homework, but please indicate clearly the names of other students you collaborated with and write your solutions separately.

Announcements

  1. On the week of February 13 the recitation and the Wednesday class will be switched: both classes will be on Monday, February 13 (one 8:00-9:15 in WWH 202, and the other as usual 3:30-4:45 in Silver 706). The recitation on the week of February 13 will be on Wednesday, February 15, 3:30-4:45 in Silver 706.
  2. The first quiz will be on February 13, in the afternoon class.
  3. The midterm will be on March 7, in class.
  4. The second quiz will be on April 11, in class.
  5. Recommended reading for the recitation meeting on January 30: Greenleaf's notes on integers, pages 7-9, from section 2.1.24 to the beginning of section 2.2.
  6. Assaf Naor's office hours will start in the second week of the semester, i.e., on February 1. Lukas Koehler's office hours will start on February 2.
  7. Suggested reading for the recitation on February 6: the GCD algorithm and example of GCD computations in Greenleaf's notes on integers pages 13-15.
  8. The final exam is scheduled for Wednesday 05/09, 4:00PM-5:50PM. See http://www.nyu.edu/registrar/pdf/Final_exam_schedule_Spring_2012.pdf for more information. Since the registrar's office sometimes changes the exam schedule please check this link again closer to the exam date.
  9. For the rest of the semester, the office hours of Lukas Koehler will be at the same time as originally announced (Mondays 5:00-6:00), but their location is moved to the seminar room 605 WWH.
  10. The office of Lukas Koehler has been permanently changed to WWH 510.
  11. Starting the week of February 13, the mechanism of homework collection will permanently change: solutions must be handed in by each Wednesday at noon, in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH). Alternatively, you can continue handing in solutions by emailing them to Lukas Koehler.
  12. The topics of the first quiz, which will take place on Februray 13 in the afternoon class, are homework 1 and the definitions and statements that we covered in class up to (and including) subgroups.
  13. The topics of the midterm, which will take place on March 7 in class, are all the material covered in class up to and including automorphisms, and homeworks 1 through 5.
  14. Note that you have three weeks to do Homework 6 (it is due after spring break), but doing it now would be a good preparation for the upcoming midterm (especially the first question on quotients of groups).
  15. Due to the upcoming midterm, Assaf Naor will hold extra office hours on Monday March 5 at 11:00-12:00 in 1113 WWH.
  16. The topics of the second quiz, which will take place on April 11 in the beginning of class, are homeworks 6 through 8 and the definitions and statements that we covered in class from Cayley's theorem up to (and including) the three Sylow theorems.
  17. The final exam will take place on Wednesday, May 9, 4:00-5:50 in class (Silver 706). The final exam will cover all the material that was covered in class this semester, as well as homeworks 1-12 (including the questions that you were not required to hand in).

Homework assignments

  1. Homework 1. Posted January 23, due February 1 (noon) in the mailbox of Lukas Koehler.
  2. Homework 2. Posted February 1, due February 8 (noon) in the mailbox of Lukas Koehler.
  3. Homework 3. Posted February 8, due February 15 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  4. Homework 4. Posted February 15, due February 22 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  5. Homework 5. Posted February 22, due February 29 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  6. Homework 6. Posted February 29, due March 21 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  7. Homework 7. Posted March 21, due March 28 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  8. Homework 8. Posted March 28, due April 4 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  9. Homework 9. Posted April 4, due April 11 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  10. Homework 10. Posted April 11, due April 18 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  11. Homework 11. Posted April 18, due April 25 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).
  12. Homework 12. Posted April 25, due May 2 (noon) in an envelope that will be posted on the door of the office of Lukas Koehler (510 WWH).

List of lecture topics

  1. Lecture 1 (January 23): Axiomatic definition of the integers and basic properties ( Greenleaf's notes on integers).
  2. Lecture 2 (January 25): Continued axiomatic definition of the integers; order axioms and induction axiom ( Greenleaf's notes on integers).
  3. Lecture 3 (January 30): The minimum property, arithmetic in the integers, Euclidean division, greatest common divisors, prime factorization (Section 1.3 in Herstein and Greenleaf's notes on integers).
  4. Lecture 4 (February 1): Conclusion of the discussion of the integers (uniqueness of prime factorization), equivalence relations (Herstein pages 6-8), modular arithmetic (Herstein pages 22-23), and definition of a group (Herstein section 2.1 and parts of Greenleaf's notes on groups).
  5. Lecture 5 (February 6): Examples of groups (Herstein sections 1.2, 2.1,2.2, and parts of Greenleaf's notes on groups).
  6. Lecture 6 (February 8): Basic properties of groups, Wilson's theorem (see http://en.wikipedia.org/wiki/Wilson's_Theorem ), subgroups (section 2.3 and parts of section 2.4 in Herstein).
  7. Lecture 7 (February 13): More on subgroups, Lagrange's theorem, flood (section 2.4 in Herstein).
  8. Lecture 8 (February 13): Applications of Lagrange's theorem, including Fermat's theorem and the counting principle for products of subgroups, beginning of the discussion of normal subroups (sections 2.4, 2.5 and the beginning of section 2.6 in Herstein).
  9. Lecture 9 (February 22): Properties and equivalent characterizations of normal subgroups, quotient groups, definition of homomorphism and isomorphism and their basic properties, examples of homomorphisms, the kernel and image of a homomorphism, statement of the first isomorphism theorem (section 2.6 in Herstein and the beginning of section 2.7 in Herstein).
  10. Lecture 10 (February 27): More on homomorphisms, proof of the first isomorphism theorem, definition of simple groups and historical discussion on the classification of finite simple groups (for more information see http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups ), Cauchy's theorem for Abelian groups (section 2.7 in Herstein).
  11. Lecture 11 (February 29): Conclusion of the proof of Cauchy's theorem for Abelian groups, Sylow's first theorem for Abelian groups, automorphisms and inner automorphisms (sections 2.7 and 2.8 in Herstein).
  12. Lecture 12 (March 5): Review for midterm exam, more on automorphisms (Lemma 2.8.3 and Example 2.8.1 in Herstein).
  13. Lecture 13 (March 7): Midterm exam.
  14. Lecture 14 (March 19): Cayley's theorem, permutation groups, orbit decomposition, product of cycles (sections 2.9, 2.10 in Herstein and Greenleaf's notes on permutation groups).
  15. Lecture 15 (March 21): Cycle type of a permutation, every permutation is a product of transpositions, sign of a permutation, the alternating group An, beginning of the classification of conjugate permutations (section 2.10 in Herstein, pages 88--89 in Herstein, and Greenleaf's notes on permutation groups).
  16. Lecture 16 (March 26): Conclusion of the classification of conjugate permutations, proof of the simplicity of An for n≥ 5 (pages 88--89 in Herstein, and Greenleaf's notes on permutation groups or http://planetmath.org/encyclopedia/SimplicityOfA_n.html).
  17. Lecture 17 (March 28): Conclusion of the proof of the simplicity of An for n≥ 5, conjugacy classes, the class equation, some applications of the class equation: the center of a group of order pn is non-trival, groups of order p2 are Abelian, Cauchy's theorem for general groups (section 2.11 in Hertstein).
  18. Lecture 18 (April 2): More applications of conjugacy classes and the class equation, proof of Sylow's first theorem, statement of Sylow's second theorem (end of section 2.11 and beginning of section 2.12 in Herstein).
  19. Lecture 19 (April 4): Proof of Sylow's second theorem and Sylow's third theorem (section 2.12 in Herstein).
  20. Lecture 20 (April 9): Conclusion of the proof of Sylow's third theorem, examples of applications of the Sylow theorems: any group of size 112132 is Abelian, classification of groups of size pq (section 2.12 in Herstein).
  21. Lecture 21 (April 11): Quiz in the first part of class. More examples of applications of the Sylow theorems: a group of size 72 cannot be simple and any simple group of size 60 is isomorphic to A5 (section 2.12 in Herstein).
  22. Lecture 22 (April 16): Conclusion of the proof that any simple group of size 60 is isomorphic to A5. Direct products and the fundamental structure theorem for finite Abelian groups (parts of sections 2.13 and 2.14 in Herstein).
  23. Lecture 23 (April 18): Definitions of a ring, field and division ring. Many examples of rings. Zero divisors and intergral domains. Finite integral domains are fields (sections 3.1 and 3.2 in Herstein).
  24. Lecture 24 (April 23): Characteristic of integral domains. Ring homomorphisms, ideals, quotients of rings by ideals, the first isomorphism theorem for rings (sections 3.2, 3.3, 3.4 in Herstein).
  25. Lecture 25 (April 25): Euclidean rings and principle ideal rings. Arithmetic and prime factorization in Euclidean rings (section 3.7 in Herstein).
  26. Lecture 26 (April 30): Polynomial rings of fields. The division algorithm for polynomials and irreducible polynomials. Proof that if F is a field then F[x] is a Euclidean ring (section 3.9 in Herstein).
  27. Lecture 27 (May 2): Proof that the Gaussian integers are a Euclidean ring, proof of Fermat's characterization of primes that are sums of two squares. Maximal ideals (section 3.8 and 3.5 in Herstein).
  28. Lecture 28 (May 7): Review for final exam.