# Math 120: Modern algebra

Fall 2008
Tuesday and Thursday 9:30-10:45 in 380-X

This class introduces basic structures in abstract algebra, notably groups, rings, and fields. Within group theory, we will discuss permutation groups, finite Abelian groups, p-groups, and the Sylow theorems. Within ring theory, we will discuss polynomial rings, principal ideal domains, and unique factorization domains. This is a Writing in the Major class.

This will be a fast-moving, high-workload class. Most students interested in this material will find Math 109 more appropriate.

Professor: Ravi Vakil, vakil@math, 383-Q, office hours Tuesday 10:45-12:45.

Course assistant: Anssi Lahtninen, lahtinen@math, 381-J, office hours Monday 10-12, Tuesday 5-7, Wednesday 10-12.

Text: Dummit and Foote's Abstract Algebra, 3rd ed (be careful to get the right edition!).

Homework 20%
Writing in the Major Assignment 20%
Midterm 20%
Final exam 40%

Homework. There will be weekly homework assignments, posted here. Solutions will be posted in this directory. You are encouraged to work together to solve problems. But you must write up your solutions individually, and give credit for ideas that others had. You should give complete proofs. Because the grader will need to process a huge volume of homework in a small amount of time: Please staple your homework, and write your name on each page.

Problem sets will be due on Thursdays at the start of class. No lates will be accepted, and instead I will drop your lowest homework score. In particular, everyone gets sick from time to time, and has exceptionally busy periods, so don't make use of this too early in the quarter!

You can hand in homework earlier in my mailbox, on the ground floor of the math department, across from the elevators. The grader in general intends to return the problem sets by the next class. No lates will be allowed (so the grader can just have to grade one problem set at a time, and hence have a better chance of getting them back promptly). But to give everyone a chance to get sick, or have busy periods, the lowest problem set will be dropped.

• Problem Set 1, due Thursday October 2 at 11 am. Do 18 of the following problems.
• 1.1 problems 9b, 22, 25, 31
• 1.2 problem 16. Here "is" means "is isomorphic to". Be sure that your proof is watertight!
• 1.3 problems 19, 20
• 1.4 problems 8
• 1.5 problem 1
• 1.6 problems 4, 9, 17, 20, 24, 26
• 1.7 problems 18, 19, 20, 23
• Show that every element of a finite group has finite order. Is the converse true?
• Problem Set 2, due Thursday October 9 at 11 am. Do 16 of the following problems.
• 2.1 problems 6, 8
• 2.2 problems 5, 7, 12
• 2.3 problems 9, 12, 20, 21, 22, 23, 26
• 2.4 problems 3, 7
• 2.5 problems 2, 4, 10
• Problem Set 3, due Thursday October 23 at 11 am. Do 16 of the following problems, including at least one from each section.
• 3.1 problem 9, 32, 35, 36, 41, 42
• 3.2 problem 4, 8, 19, 22, 23
• 3.3 problems 3, 9
• 3.4 problem 1
• 3.5 problems 3, 10
• 4.1 problems 1, 10 (counts for 2 problems)
• Problem Set 4, due Thursday October 30 at 11 am. Do 15 of the following problems, including at least one from each section.
• 4.2 problems 1, 8, 9, 10
• 4.3 problems 4, 5, 21, 28, 29, 31
• 4.4 problems 1, 2, 8, 18 (counts for two), 19
• 4.5 problems 7, 13, 29
• Problem Set 5, due Thursday November 6 at 11 am. Do 10 of the following problems, including at least one from each section. (The writing in the major project draft is due on Nov. 7.)
• 4.5 problem 3
• 4.6 problems 1, 2
• 5.1 problems 1, 4, 11
• 5.2 problems 1, 5, 9
• 5.4 problems 2, 4, 5
• 5.5 problems 1, 2, first sentence of 8
• Problem Set 6, due Thursday November 13 at 11 am. Do 12 of the following problems, including at least one from each of the four sections of the book.
• 7.1 problems 6, 14, 23, 25
• 7.2 problems 3, 10, 13
• 7.3 problems 13, 24, 29, 34
• 7.4 problems 4, 5, 11. You will have to look up the definitions of maximal and prime ideals.
• Show that the kernel of the map Q[x] --> R given by x --> sqrt{2} is the principal ideal generated by x^2-2.
• Show that the kernel of a ring homomorphism is an ideal (in your own words!).
• Problem Set 7, due Thursday November 20 at 11 am. Do 13 of the following problems, including at least one from each section.
• 7.4 problems 10, 15, 30, 33
• 7.5 problem 3
• 7.6 problem 7
• 8.1 problems 1c, 2c, 3, 6, first sentence of 7, 8a, 10, 12
• 8.2 problems 1, 5
• 8.3 problem 2
• Problem Set 8, due Thursday December 4 at 11 am. This is an extra "greatest hits" problem set, so I'll drop the lowest two problem sets. Do 16 of the following problems.
• 1.7 problem 21
• 2.2 problem 13
• 2.4 problems 12, 17
• 3.1 problem 34
• 3.2 problems 10, 17, 21
• 3.3 problem 10
• 3.4 problem 5, 9
• 3.5 problem 12
• 4.2 problem 13 (don't just quote earlier exercises)
• 4.3 problem 33
• 4.5 problem 30
• 4.6 problem 5
• 5.1 problem 18
• 5.2 problem 14
• 5.4 problem 15
• 5.5 problem 5(b)
• 7.1 problem 26
• 7.2 problem 5
• 7.3 problem 26
• 7.4: show that there exists a field with 9 elements. (Hint: 7.4 problem 15).
• 7.5 problem 5
• 8.1 problem 9
• 8.2 problem 8
• 8.3 problem 7

Writing in the major assignment. This course will emphasize both exposition in communciating mathematics and the structure of proofs. Part of your grade on each assignment and on the exams will be on your exposition of your solutions to problems. There is be a writing assignment, worth 20% of the grade. Information about it is here. Deadlines to be aware of: Nov. 7 and Nov. 21.

Approximate syllabus. You should read the "preliminaries" part of the book before the class starts.

Sept. 23: (chapter 1: introduction to groups) What this course is about, and why groups are important. Binary operation, associative operation, commutative operation, group, abelian group, commutative group, finite group, dihedral groups.
Sept. 25: fields, F_p, GL_n(F), SL_n(F), symmetric groups S_I and S_n, cycle notation, homomorphisms and isomorphisms of groups, group actions.
Sept. 30: (chapter 2: subgroups) definition, stabilizer, kernel, centralizer, normalizer.
Oct. 2: kernel, image, cyclic (sub)groups, subgroups generated by subsets, lattice of subgroups of a group. (chapter 3: quotients and homomorphisms) equivalence relations, left (and right) cosets.
Oct. 7: chapter 3: quotients and homomorphisms, left and right cosets, Lagrange's theorem, index, classification of groups of order p, normal subgroup, quotient group, |HK|=|H| |K| / |H cap K|.
Oct. 9: isomorphism theorems, composition series, Holder program, simple group, Jordan-Holder theorem, solvable group, transpositions and the alternating group.
Oct. 14: chapter 4: group actions: permutation representation; kernel, fathful, orbit, bijection between left cosets and elements of orbit, groups acting on themselves by left-multiplication, Cayley's theorem, groups acting on themseles by conjugation, class equation, p-groups have nontrivial center.
Here is a practice midterm.
Oct. 16: Midterm in class. The median was 20. (The drop deadline is Sunday. If you would like to know how the midterm went before then, please send me an e-mail, and I'll respond by Saturday night.) Here are the solutions, by Anssi.
Oct. 21: chapter 4 continued: conjugacy in S_n, A_5 is simple, automorphisms of groups, statement of the Sylow theorems.
Oct. 23: consequences of Sylow; classification of simple groups of order less than 60; beginning of proof.
Oct. 28: proof of Sylow theorems; A_n is simple; direct products, and how to recognize them.
Oct. 30 (lecture by Prof. Kannan Soundararajan): fundamental theorem of finitely generated abelian groups; semidirect products; defining a semidirect product given a map from one group to the automorphisms of another.
Nov. 4: rings; bells and whistles (commutative; multiplicative identity 1; 1 not 0; field); zero-divisors, units, integral domain, subring, ring homomorphisms.
Nov. 6: ideals (e.g. ideals of Z or F[x]), quotient of a ring by an ideal, four isomorphism theorems for rings
Fri. Nov. 7: draft of writing assignment due.
Nov. 11: chapter 7 continued, and chapter 8: chinese remainder theorem, principal ideals.
Nov. 13: euclidean algorithm, every principal ideal domain is a unique factorization domain.
Nov. 18 and Nov. 20 canceled due to family illness. Please read until the end of 8.3, including factorization of the Gaussian integers!
Fri. Nov. 21: writing in the major project due.
Dec. 2: toward Galois theory: the impossibility of trisection and doubling the cube.
Dec. 4: review.
Monday, December 8: Final exam, 12:15-3:15 in 380-C. Here is a practice final.