This class will cover groups, fields, rings, and ideals. More explicitly: Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains.

Math 120 will be a fast-moving, high-workload class. Most students
interested in this material will find Math 109 more appropriate.
This is also a
*Writing in the Major* class.

**Professor:** Ravi Vakil, vakil@math, 383-Q,
office hours (chosen by popular demand) Wednesday afternoon 2-2:30 and 3:30-5.

**Course assistant:** Amy Pang, 381-F, amypang@stanford.
Office hours: Monday 1:30-3:30, Tuesday
9:30-10:30, Thursday 9:15-10:45 and 4-5:30.

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

**Grading scheme**

Short online assignments 5%

Problem sets 20%

Writing in the Major Assignment 15%

Midterm 20%

Final exam 40%

**Short online assignments.**
There will be weekly short online assignments on Coursework.
You needn't answer more than one or two sentences per question.
This is intended to be low-stress: you will get full marks for any
good-faith answer.
This is intended to give us feedback on how the reading has been
going. Because I will have to review your feedback in a limited time
period on Sunday afternoon, the deadlines will be on Sunday at
noon sharp (no submissions will be accepted after that).

**Problem sets.** There will
be weekly homework assignments, posted here.
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.

Homework is due in class on Friday, but you can also hand it
in by 1 pm in my mailbox at the math department (first floor, across
from the elevator). (If you are
intending to hand it in after class in my mailbox, make sure you know
in advance where my mailbox is!)
**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 Friday Oct. 2: Do 15 of the following 19 problems. Section 1.1: problems 1, 7, 9, 22, 26, 28. Section 1.2: problem 5, 11. Section 1.3: problems 8, 20. Section 1.4: problem 1. Section 1.6: problems 4, 14, 17. Section 1.7: problems 3, 4, 18, 19, 21.
- Problem set 2, due Friday Oct. 9: Do 14 of the following problems. Section 2.1: problems 1, 6, 8, 12, 16. Section 2.2: problems 2, 5, 7, 12 (counts for 2). Section 2.3: problems 1, 12, 19, 21, 26. Section 2.4: problems 7, 8. Section 2.5: problems 2, 4.
- Problem set 3, due Friday Oct. 16: Do 13 of the following problems, including at least one from each section. Section 3.1: 1, 9, 22, 24, 31, 41. Section 3.2: 4, 10, 11, 17, 22. Section 3.3: 3, 7, 9. Section 3.4: 4, 5. Section 3.5: 10, 12. Misc.: Show that the rotation group of the sphere is generated by the rotations around the north pole and the rotations around the "east pole". You may use the fact any rotation of the sphere is a rotation around some axis. (Challenge: your write-up must be convincing to the grader.)
- Problem set 4, due Friday Oct. 23: Do 13 of the following problems, including at least one from each section. Section 4.1: 2, 4, 10. Section 4.2: 2, 8, 10. Section 4.3: 4, 5, 17, 25, 29, 33. Section 4.4: 1, 2, 6, 8, 16.
- Problem set 5, due Friday Nov. 6: Do 13 of the following problems, including at least one per section. Section 4.5: 6, 30, 33, 42. Section 4.6: 1, 2, 4, 6. Section 5.1: 9 (no need use the previous exercise unless you want to), 11, 18 (do 4 out of 5 parts). Section 5.2: 1, 5, 8, 9, 12. Section 5.5: 2, 4.
- Problem set 6, due Friday Nov. 13: Do 13 of the following problems, including at least one per section. Setion 5.4: 2, 4, 5. Seciton 6.1: 5, 6, 14, 16. 7.1: 5, 7, 8, 11, 13, 21, 23, 25, 26.
- Problem set 7, due Friday Nov. 20: Do 13 of the following problems. 7.2: 3, 12. 7.3: 2, 10, 18, 22, 25, 29, 34. 7.4: 9, 11, 27, 30, 34, 37, 41.
- Problem set 8, due Wednesday Dec. 2: Do 10 of the following problems, including at least one from each section. 7.5: 2, 5. 7.6: 6, 7. 8.1: 6, 7, 10, 12. 8.2: 1, 5. 8.3: 1, 4, 5, 6.

**Writing in the major assignment.** Clear writing is essential
to mathematical communication, as you probably realize from reading
better and worse mathematical texts. Clarity in writing may be more
important in mathematics than in any other science, for a number of
reasons. Good exposition is an acquired and important
skill. Throughout this class, you'll received feedback on solutions to
your problem sets, and you should use this to refine your ability to
communicate your ideas clearly and effectively. This writing project
will give you an opportunity to focus on your exposition, as opposed
to absorbing new mathematical content. 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. Information
about the Writing in the Major assignment is here.
Deadlines to be aware of: October 30 and November 30.

**Final Exam:**
The final exam will take place on Friday, December 11 from 8:30 to
11:30 am in 380Y (in the basement of the math department). There will be two review sessions (or bonus office hours,
depending
on what people want) beforehand. The first will be run by Amy on
Monday December 7 from 10:30 am to 1:30 pm. The second will be run by me on
Thursday December 10 from 2 to 5 pm. They will take place in our
respective offices.
Here is
a practice final exam.

**Syllabus.**

Sept. 21: Welcome. Groups: definition and examples.

Sept. 23: Dihedral groups (generators and relations; normal form), cyclic groups, symmetric groups, matrix groups, fields. Inklings of group isomorphisms and homomorphisms, and group actions. The locker puzzle.

Sept. 25: Homomorphisms and isomorphisms. What can you say about order of elements given the order (size) of the group? Group actions.

Sept. 28: Isomorphism questions, subgroups, group actions. Implicit ideas: stabilizers, subgroups generated by a subset, cyclic subgroups.

Sept. 30: More subgroups, stabilizer subgroup, kernel of a group action, kernel of a group homomorphism, centralizer of a subset, normalizer. Implicit ideas: quotient groups, cosets, normal subgroups.

Oct. 2: subgroups generated by a subset (definitions, examples of proofs), lattices of subgroups of a group.

Oct. 5: quotient group examples, cosets, Lagrange's theorem, Cauchy's theorem, odd permutations.

Oct. 7: quotient groups, normal subgroups (many equivalent definitions, isomorphism theorems.

Oct. 9 (last day to add or drop): |HK| = |H| |K| / |H cap K| and variations; more isomorphism theorems; composition series and the Jordan-Holder theorem; solvable groups; the alternating group.

Oct. 12: kernel, stabilizer, faithful, transitive. Orbits vs. fibers vs. equivalence classes. Kernel of G-action on G/H is intersection (over x in G) of x H x^{-1}. Applications: Cayley's theorem, and normality of index p subgroups of groups of size n (where p is the smallest prime dividing n). Better version of "size of orbit is index of stabilizer": there is a natural bijection between the orbit of a and the stabilizer of a, which is ga <--> g Stab(a).

Oct. 14: groups acting on themselves by conjugation, and the class equation.

Oct. 16: automorphisms.

Oct. 19: Sylow theorems: statement, applications, start of the proof.

Oct. 21: Proof of Sylow theorems.

Oct. 23: Review of the course so far. The last 20 minutes will be used for a Small Group Evaluation of the class by Robyn Dunbar of the Center for Teaching and Learning.

Oct. 26: Midterm. Here is a practice midterm. Here is something that may give you an entertaining break while studying (thanks to Charlotte). Amy is willing to answer more questions on Friday, and Monday before the midterm; just email her to set an appointment. Here are solutions to the midterm.

Oct. 28: simplicity, and simplicity of the alternating group on n letters (n sufficiently large).

Oct. 30: (Look over 5.1 and 5.2.) Writing in the Major draft due (handed in to Tracy Nance's mailbox by noon, or in class).

Nov. 2: Relationship between the two versions of the fundamental theorem of finite(ly generated) abelian groups. Why we care about nilpotence. p-groups are nice. A nilpotent finite group is a product of its Sylows.

Nov. 4: The commutator subgroup, and its "descendants": the commutator series (and solvability) and the lower central series (and nilpotence). The start of the proof of the fundamental theorem of finite abelian groups: part 1 (G is a product of p-groups), and part of part 3 (uniqueness).

Nov. 6: Proof of the classification (=fundamental) theorem of finite abelian groups; semidirect products.

Nov. 9: introduction to rings (e.g. commutative, with unit, division ring, field, integral domain). Zero-divisor, subring. Examples. Ring homomorphisms, isomorphism, kernel, image. Ideals and their motivation. The quotient ring.

Nov. 11: Properties of rings descend to their quotients. Learning more about quotients by ideals through examples.

Nov. 13: The four isomorphism theorems (for rings); maximal and prime ideals, and quotients by them. Change of grading basis deadline. Course withdrawal deadline.

Nov. 16: fractions, the Chinese remainder theorem, euclidean domain, euclidean algorithm.

Nov. 18: the ideals of Z; euclidean domains must be principal ideal domains; principal ideal domains must be unique factorization domains.

Nov. 20: variations on Fermat's last theorem (inspired by the talk in today's number theory seminar), and why you should be nervous about the axiom of choice.

Nov. 30: Fermat's two-square theorem, and the primes of Z[i]. Writing in the Major assignment due (in Tracy Nance's mailbox by noon, or in class).

Dec. 2: Impossibility of trisection and doubling the cube.

Dec. 4: Review of quarter.

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