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Lectures:

MWF 9:05 a.m. - 9:55 a.m. in Vincent Hall 113

Prerequisites:

Math 2283 or 3283

Instructor:

Christine Berkesch Zamaere

Office: Vincent Hall 250

Email: cberkesc -at- math.umn.edu (For faster response time, please include "Math4281" in the subject line.)

Office hours:

Monday 10:00 a.m. - 11:30 a.m.,

Friday 10:00 a.m. - 10:30 a.m.,

or by appointment (in Vincent Hall 250).

Textbooks:

Reading a book will help supplement your class notes; however, you need not read both.

Abstract Algebra: Theory & Applications by Thomas Judson, 2013 ed. (Available free online at the given link.)

Abstract Algebra: a geometric approach by Theodore Shifrin, 1996. (On reserve in the Mathematics Library, Vincent Hall 310.) Errata and typos here.

Description:

This is an introductory course in modern algebra. It differs from Math 5285H: Fundamental Structures of Algebra by being less theoretical and having somewhat different subject matter (although there is some overlap).

Getting help:

• You are strongly encouraged to work in study groups and learn from each other, although all final work submitted must be your own. Lind Hall 150 is available for small group meetings and individual study.
• Free tutoring services are available. See Tutoring Resources at the Undergraduate Math page for more information.
• Hire a tutor. A list of tutors is available in Vincent Hall 115 or by email request at ugrad@math.umn.edu.
• Attend my office hours (see above).
• Email me (above) short questions or comments. Please allow at least one working day for a response.

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Staple the appropriate problem set sheet to the front of your assignment.

Assignment		Due date		Quiz date
Problem Set 1		Wednesday, January 29		Friday, January 31
Problem Set 2		Wednesday, February 5		Friday, February 7
Problem Set 3		Wednesday, February 12		Friday, February 14
Problem Set 4		Wednesday, February 19		Friday, February 21
Problem Set 5		Wednesday, February 26		Friday, February 28
Problem Set 6		Wednesday, March 5		Midterm 1 on Friday, March 7, over all sections through "Irreducible polynomials"
Problem Set 7		Wednesday, March 12		Friday, March 14
Problem Set 8		Wednesday, March 26		Friday, March 28
Problem Set 9		Wednesday, April 2		Friday, April 4
Problem Set 10		Wednesday, April 9		Friday, April 11
Problem Set 11		Wednesday, April 16		Friday, April 18
Problem Set 12		Wednesday, April 23		Midterm 2 on Friday, April 25, over all sections from "Ring homomorphisms and ideals" through "Normal subgroups and quotient groups"
Problem Set 13		Wednesday, May 7		Friday, May 9

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Below is a list of topics and corresponding reading for the semester, which is subject to change. As they become available, I will add the lecture dates. No topic next to a date means that we will continue with the previous topic.

Date		Topic		Shifrin		Judson
01-22		Preliminaries		A.1, A.2		1.1, 1.2 thru p. 14
01-24		Properties of the integers		1.1		2.1
01-24		Division and Euclidean algorithms		1.2		2.2
01-27		All University classes cancelled
01-29
01-31
02-03		Modular arithmetic		1.3		3.1 thru top of p. 40, 6.3 (skip Theorems 6.11 and 6.12)
02-03		Solving congruences		1.3		16.5
02-05		Equivalence relations		A.3		1.2
02-07		Rings, domains, and fields		1.4		16.1 (ignore division rings, we will focus on rings with 1), 16.2 (only a special case of Theorem 16.4 to Theorem 16.6, see class notes)
02-10
02-12		The complex numbers		2.3		4.2 (skip Prop. 4.10)
02-14		Introduction to polynomials		3.1		17.1 (skip Theorem 17.3)
02-17		Euclidean algorithm for polynomials		3.1		17.2
02-19		Roots of polynomials		3.2		21.1 (stop after Example 1), 21.2 (stop after Example 11)
02-21
02-24		Irreducible polynomials		3.3		17.3 (stop after Example 7)
02-26
02-28		Ring homomorphisms and ideals		4.1		16.3, Theorem 17.3
03-03
03-05		Review for Midterm 1
03-07		Midterm 1				(over all sections of notes through "Irreducible polynomials")
03-10		Quotient Rings		4.1		16.3
03-12		Ring isomorphisms		4.2		16.3, 21.1 through Theorem 21.2, 21.2 through Theorem 21.17
03-14
03-26
03-26		Vector spaces and field extensions		5.1		20.1, 20.2, 20.3, 21.1 after Theorem 21.3 through Example 9
03-28
03-31		Groups		6.1		3.1, 3.2, 3.3
04-02		Cyclic groups		6.1		4.1
04-04
04-07		Permutation groups		6.4		5.1, 5.2
04-09		Group homomorphisms and isomorphisms		6.2		11.1 (skip Theorem 11.2), 9.1 up to Theorem 9.5
04-11
04-14		Cosets		6.3		6.1, 6.2
04-16		Normal subgroups and quotient groups		6.3		10.1 (Note that there are several equivalent definitions of normal subgroups!), Theorem 11.2, 11.2 through Example 8
04-18
04-21
04-23		Review for Midterm 2
04-25		Midterm 2				(over all sections from "Ring homomorphisms and ideals" through "Normal subgroups and quotient groups")
04-28		Galois theory		7.6		21.2, 23.1, 23.2, 23.3
05-02
04-30
05-05		History of solving polynomials		N/A		Handout
05-07		Review for Final on May 15
05-09

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January 22		First Math 4281 class meeting
January 28		Last day to register without instructor approval and drop with a 100% refund
February 3		Last day to drop without receiving a "W" and with a 75% refund
March 7		Midterm 1
March 17-21		Spring break
April 25		Midterm 2
May 9		Last day of class
May 15 (Thursday)		Final exam, 1:30 p.m. - 3:30 p.m.

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Homework:

Weekly problem sets are posted under homework above. Problem sets will be collected in class on Wednesday (unless otherwise announced) and will cover roughly the material from the previous week. Late homework will not be accepted, but early hard copy submission is welcomed. If you have an unavoidable and legitimate university sanctioned excuse for missing an assignment, please contact me as soon as possible about this issue. Your two lowest problem sets will be dropped from your grade.

The homework in this course is intended to be challenging; it is training to help you build stronger mathematical muscles. There will be problems whose solutions are not immediate; there will be times that you will even need to sleep on a problem before fully grasping its solution. With this in mind, I recommend that you give yourself a full week to do the homework, so that only a few challenges remain by the Monday before the assignment is due. This gives you ample time to discuss any difficulties with your classmates and attend office hours in order to complete the problem set.

While you are encouraged to consult with your classmates on the homework, your final work must be your own. Copying a classmate's work constitutes plagiarism and violates the University of Minnesota Student Conduct Code. If you collaborate to reach an answer, include the name of your classmate(s) with your solution.

Selected homework problems (or similar) may be given on quizzes and exams. This is another reason why you should do the homework before each class and, moreover, remember the ideas and techniques used in your solutions.

Quizzes:

There will be weekly quizzes related to the previous homework assignment and its relevant definitions and theorems. These will typically be at the beginning of class each Friday. Make-up quizzes will not be given, but you may take a quiz early. Your two lowest quiz scores will be dropped from your grade. All quizzes are closed book, closed notes, and no calculators.

Midterm and Final Exams:

There will be two midterms in class, on March 7 and April 25. The final exam will be on Thursday, May 15, from 1:30 p.m. - 3:30 p.m. All exams are closed book, closed notes, and no calculators.

Grading:

Your grade is based on homework, quizzes, and exams, which will be weighted as follows:

Homework		30%
Quizzes		10% total
Midterm exams (x2)		17% each
Final exam		26%

Missed exams:

No make-up quizzes or exams will be given; however, it is possible to take a quiz or exam early if you have a valid reason. If you have an unavoidable and legitimate university sanctioned excuse for missing an exam, please contact me as soon as possible about this issue.

Written work:

We write to communicate. Please keep this in mind as you complete written work for this course. Work must be neat and legible in order to receive consideration. You must explain your work in order to obtain full credit; an assertion is not an answer. The logic of a proof must be completely clear in order to receive full credit.

Reading:

You will find the lectures easier to follow if you spend time with your textbook before class. The lectures section will tell you the topics for the coming class meetings. Before class, skip proofs, but seek to understand the "big idea" of each section, the key definitions, and statements of the main theorems. After class, read all statements and proofs carefully, and stop to identify useful proof techniques along the way.

Definitions and Theorems:

In order to be successful in this course, it is imperative that you memorize definitions and important theorem statements. I suggest having a special, separate place in your notebook to record this information. Flashcards might also be helpful.

Workload:

You should expect to spend about 9 hours a week outside of class on reading and homework for this course. This course builds upon itself, so in order to be successful, it is important to not fall behind.

Technology:

Students are encouraged to use technology available to them for homework, but no technical aids will be allowed on the exam.

Disabilities:

Students with disabilities, who will be taking this course and may need disability-related accommodations, are encouraged to make an appointment with me as soon as possible. Also, please contact U of M's Disability Services to register for support.

Academic integrity:

It is the obligation of each student to uphold the University of Minnesota Student Conduct Code regarding academic integrity. You will be asked to indicate this on your homework assignments. Students are strongly encouraged to discuss the homework problems but should write up the solutions individually. Students should acknowledge the assistance of any books, software, students, or professors.