## Math 150B - Algebra Winter 2013

Instructor: Brian Osserman

Lectures: MWF 3:10-4:00, Hoagland 168.

CRN: 59891

Office: MSB 3218, e-mail:

Office Hours: M 4:10-5:00, F 1:10-2:00

Prerequisites: Math 150A

Textbook: Michael Artin, Algebra (2nd edition)

Syllabus: The main topics for the quarter will be bilinear forms, linear groups, and ring theory. See also the department syllabus.

TA: Binglin Li

Discussion: T 3:10-4:00, Hoagland 168

TA Office Hours: T 12:10-1:00, MSB 3131

Grading: 30% homework, 25% midterm, 45% final exam

Homework: Assigned weekly, due each Wednesday in class

### Welcome to Math 150B: Algebra

Algebra is a core branch of mathematics. It is important in and of itself, and also to a wide range of other fields, including number theory, algebraic geometry, and algebraic topology. In addition, it has applications ranging from cryptography to crystallography. We will begin with bilinear forms and linear groups, and then spend the majority of the course on the theory of rings, including ideals, factorization, and modules.

### Lecture notes

I will post notes here to supplement Artin as seems appropriate.

• Sesquilinear forms: This is an alternative presentation of the material we covered from sections 8.1 - 8.5 and 8.8 of Artin, organized to minimize redundancy.
• Summary: linear algebra and linear groups: This summarizes all the definitions, and the key results, from the first four weeks of the course.
• Polynomial rings and unique factorization domains: This is the more general version of Section 12.3 of Artin which was presented in lecture.
• Summary: rings: This summarizes all the definitions, and the key results, from the material on rings in Chapters 11-13 of Artin.
• Summary: modules: This summarizes all the definitions, and the key results, from the material on modules in Chapter 14 of Artin.

### Problem sets

Problem sets will be posted here each Wednesday, due the following Wednesday in class. You are encouraged to collaborate with other students, as long as you make sure you understand your answers and they are in your own words. You are not, under any circumstances, allowed to get answers to problems from any outside sources.

A selection of problems will be graded from each problem set. To minimize resulting randomness of scores, your lowest problem set score will be dropped when calculating your grade.

• Problem set #1, due 1/16: do Exercises 3.2.1, 3.2.11, 3.3.1, 3.3.2, 3.4.1, 3.4.5, and 4.1.2.
• Problem set #2, due 1/23: do Exercises 8.1.1, 8.3.1, 8.3.6, 8.4.1, 8.4.4, 8.4.5, and 8.4.11. (Notice the typo in 8.3.1 (at least, in my book) that it says "for all Xa", but means to say "for all X a")
• Problem set #3, due 1/30: do Exercises 8.2.2, 8.3.2, 8.4.9, 8.4.15, and 8.5.2.
• Problem set #4, due 2/6: do Exercises 8.6.1, 8.6.5, 8.6.11, 8.6.13, 9.1.1, 9.1.5, and 9.3.1.
• Problem set #5, due 2/13: do Exercises 9.3.3, 9.4.2, 11.1.7, 11.1.9, 11.3.2, and 11.3.5.
• Problem set #6, due 2/20: do Exercises 11.3.3, 11.3.8 (see the end of Section 11.3 for the definition of characteristic of a ring), 11.3.9, 11.3.11 (prove or disprove each direction separately), and 11.4.1.
• Problem set #7, due 2/27: do Exercises 11.5.2, 11.5.6, 11.7.1, 11.8.1, and 11.8.2.

• Problem set #8, due 3/6: do Exercises 12.2.3, 12.2.4, 12.2.6, 12.2.7, 12.2.10, 12.3.1 and 12.5.10.
• Problem set #9, due 3/13: do Exercises 13.5.1 (in (b), δ should be the square root of -5), 13.6.2 (R is the ring of integers in the field obtained from Q by adjoining the square root of d), 14.1.1, 14.1.2, 14.2.2, 14.2.3 (a) and 14.4.6.
• Problem set #10, optional (not to be turned in): do Exercises 14.5.1, 14.5.2, 14.7.2, 14.7.6 and 14.7.9.

### Exams

There will be one in-class midterm exam, on Wednesday, February 13. It will cover all material up through and including the lecture on February 6, corresponding to the first five homeworks. Look under Lecture Notes above for a summary of the first four weeks of material. Beyond the material in the summary, the exam will cover material from sections 11.1, 11.2, and (part of) 11.3 of Artin.

The final exam is scheduled for Friday, March 22 1:00-3:00 PM. It will cover all material from the three summaries posted above (corresponding to all 10 homeworks), but will focus on the last two.

Both exams will be held in the regular lecture room.

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