## Math 150C - Algebra Spring 2015

Instructor: Brian Osserman

Lectures: MWF 10:00-10:50, Olson 223.

CRN: 39518

Office: MSB 3218, e-mail:

Office Hours: T 2:10-3:00, W 11:00-11:50

Prerequisites: Math 150AB

Textbook: Michael Artin, Algebra (2nd edition)

Syllabus: The main topics for the quarter will be ring theory and field theory. Note that, in coordination with this year's 150B, this deviates from the usual department syllabus. The ring theory is roughly weeks 6-10 of the 150B department syllabus, while the field theory is roughly weeks 5-10 of the 150C department syllabus.

TA: Nathaniel Gallup

Discussion: R 7:10-8:00, Cruess 107

TA Office Hours: M 3:10-4:00, R 4:10-5:00, MSB 3217

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

Homework: Assigned weekly, due each Friday in class

### Welcome to Math 150C: 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 spend the first half of the quarter on the theory of rings, including factorization and modules. In the second half of the quarter, we will discuss fields and field extensions, with applications to topics such as ruler and compass constructions.

### Lecture notes

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

• Summary of rings material from 150B: this is a summary of the material on rings which I expect you to already be familiar with from the end of 150B.
• Unique factorization domains: this is a presentation of the material from Artin's Section 12.3, rephrased for polynomials over general unique factorization domains (rather than just over the integers).
• Review of definitions for midterm: A summary of all the definitions you will be expected to know for the midterm, as well as some of the basic results relating the definitions to one another.
• Splitting fields: this is the presentation of splitting fields as given in lecture.
• Finite fields: this is the presentation of finite fields as given in lecture.
• Galois theory: this is the presentation of Galois theory as given in lecture. It will be updated further as more is covered.

### Problem sets

Problem sets will be posted here each Friday, due the following Friday 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, and some points will be assigned based on the number of problems completed. To minimize resulting randomness of scores, your lowest problem set score will be dropped when calculating your grade.

• Problem set #0, "due" 4/3: do Exercises 1.6 (a), 2.1, 3.2, and 4.1 of Chapter 11. Also do Exercises 3.3 and 3.4 if you didn't do them last quarter.
Note: these are suggested review problems for last quarter's material, and are not actually to be turned in.
• Problem set #1, due 4/10: do Exercises 11.1.3, 11.3.5(b) (take as the definition of α being a multiple root that f is a multiple of (x-α)2), 11.3.9, 11.3.11 (prove or disprove each direction separately), 11.4.2, 11.5.3 and 11.5.5.
Grading: 10 points for 11.3.5(b), 10 points for 11.5.3, 10 points for completeness of remaining problems.
• Problem set #2, due 4/17: do Exercises 11.5.6, 11.5.7, 11.7.1, 11.7.2, 11.7.3, 11.8.1 and 11.8.3.
Grading: 10 points for 11.5.6, 6 points for 11.7.1, 6 points for 11.7.2, and 8 points for completeness of remaining problems.
• Problem set #3, due 4/24: do Exercises 12.2.1, 12.2.3, 12.2.5, 12.2.6, 12.3.1, 12.3.2 and 12.3.4.
Grading: 10 points for 12.2.1, 10 points for 12.3.1, 10 points for completeness of remaining problems.
• Problem set #4, due 5/1: do Exercises 12.3.6, 12.5.1, 12.5.5, 13.1.3, 13.2.1, 13.4.3, and 13.1.2.
Grading: 10 points for 12.3.6, 10 points for 13.2.1, 10 points for completeness of remaining problems.
• Problem set #5, due 5/8: do Exercises 13.1.4, 13.6.6, 15.1.1, 15.2.2, 15.3.1, 15.3.7, and 15.3.10.
Grading: 10 points for 15.2.2, 10 points for 15.3.7, 10 points for completeness of remaining problems.
• Problem set #6, due 5/15: do Exercises 15.1.2, 15.2.1, 15.2.3, 15.3.2, 15.3.6, 15.3.9, and 15.6.1.
Grading: 10 points for 15.2.3, 10 points for 15.3.2, 10 points for completeness of remaining problems.
• Problem set #7, due 5/22: do Exercises 16.3.1, 16.3.2, 16.3.3, 15.M.4, 15.7.1, 15.7.10 (hint: show that if F has pr elements, then the Frobenius map sending each element to its pth power is surjective).
Grading: 10 points for 16.3.2, 10 points for 15.7.10, 10 points for completeness of remaining problems.
• Problem set #8, due 5/29: do Exercises 15.7.5, 15.7.7, 15.7.9, 15.M.1, 16.4.1 (the first part of (a) was already done in class), 16.6.2, and 16.7.1.
Grading: 6 points for 15.7.7, 8 points for 16.4.1, 8 points for 16.7.1, and 8 points for completeness of remaining problems.
• Problem set #9, due 6/5 (you may turn in the assignment in the reader box marked for the class on the first floor of MSB): do Exercises 16.5.1 (you should assume these automorphisms keep C fixed, otherwise they are not uniquely determined by Artin's description), 16.5.3, 16.6.3, 16.7.2, 16.7.6 and 16.7.8.
Grading: 10 points for 16.5.1, 10 points for 16.7.6, and 10 points for completeness of remaining problems.

### Exams

There will be one in-class midterm exam, on Wednesday, May 6. It will cover all material up through and including the lecture on May 1, corresponding to the first five problem sets.

The final exam is scheduled for Monday, June 8 6:00-8:00 PM.

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