MATH 447-01 Abstract Algebra
Syllabus for Spring 2019

Mon & Fri, 8:00 – 8:50am – 326 Gildemeister Hall
Wed, 8:00 – 9:50am – 324 Gildemeister Hall

Instructor: Dr. Eric Errthum             Winona Email Username: eerrthum            Office: 205 Gildemeister Hall     Office Hours:  See homepage. Or by appointment on any day.

Text:      “A Gentle Introduction to Abstract Algebra” by B.A. Sethuraman
A pdf version of this text can be found on Brightspace or here: http://www.csun.edu/~asethura/GIAAFILES/GIAAV1.0/GIAAV1.0.pdf

Prerequisite: MATH327: Foundations of Mathematics. (MATH347: Number Theory is also highly recommended.)

Course Website:     http://course1.winona.edu/eerrthum/math447 

About This Course:  This course is designed to explore and classify abstract, algebraic structures through rigorous proof as well as provide a solid foundation on which to begin a study in the field of Algebra.

Nobody needs proofs in order to get something done, but understanding the theorems and their proofs will make anyone think more deeply about the domain, become better at problem solving, and feel more comfortable with the meaning - and limitations - of the practical procedures.

-- Alon Amit (http://qr.ae/TUNYYw)

 

Expectations: Students who complete this course with a passing grade are expected to be able to demonstrate the following skills: (i) Mastery of prerequisite material (discrete math, basic proof writing, etc.), (ii) Memorize and use new definitions, (iii) Write algebra proofs, (iv) Explain the relevant conceptual ideas from algebra, (v) Communicate mathematical reasoning clearly in both an oral and written format.

Seems to me the goal of undergraduate math is to provide experience in concept formation (finding meaning in abstract definitions), a wide variety of examples of structures, relationships, and approaches to proof, and generally develop mathematical thinking. The actual content is less important than suitability of the topic for elementary development.

Remember that traditional undergraduate math is not a professional degree in that it is not intended to provide information needed for specific jobs. Not even graduate study. The expectation is that beginning graduate students should be able to pick basic stuff up quickly, not that they already know it. Non-academic employers of people with math degrees have the same expectation.

Questions about whether specific content is really needed downstream miss the point.

 -- Frank Quinn (https://mathoverflow.net/q/25916)

 

Grading:        Oral Homework (scaled as needed)                         150 points------- 17.6%
                        Written Homework (scaled as needed)                    100 points------- 11.8%
                        Memory Quizzes (scaled as needed)                       150 points------- 17.6%
                        Relevancy Presentations (scaled as needed)             50 points-------- 5.9%
                        Chapter Exams (5 @ 50 points)                              250 points------- 29.4%
                        Comprehensive Final (oral & written)                    150 points------- 17.6%
                                                                                                     --------
                                                                                                         850 points total

Grades:  A = 90% (765 pts), B = 75% (638 pts), C = 60% (510 pts), D = 50% (425 pts). There will be no curving of individual assessments.

Homework:    Homework will be completed in two ways:

Oral Homework: According to the approximate schedule below, on Oral Homework days students will be chosen randomly to present individual solutions to problems from the homework assigned since the last homework day. For each problem presented, students will be graded a 0, 1, or 2 out of 2 corresponding to their level of preparedness (not necessarily correctness). When presenting a solution, you should be prepared to answer questions clarifying your work. It is not acceptable to write out a whole solution, but then when asked about a particular step to say “I don’t know.” To me this indicates you copied the homework from someone else without understanding it. If you miss an Oral Homework day, you will need to present on problems during office hours to make up the lost points.

Written Homework: The written portion of the homework is due the period after a Homework Day. This work should be typed in LaTeX, stapled, and presented in order. Each problem will be graded 0 – 2 on its level of mathematical correctness and 0 – 2 on how well-written it is. The written homework problems will be a subset of the oral homework problems. (If you don’t know about LaTeX, start here or just ask the seniors to help you out.)

http://37.media.tumblr.com/923ea3d8c6d2d43831998571b03b36f6/tumblr_mlyrq3ODiZ1rvnpe0o1_500.png

Don't just read it; fight it!

Ask your own questions, look for your own examples, discover your own proofs.

Is the hypothesis necessary? Is the converse true?

What happens in the classical special case?

What about the degenerate cases?

Where does the proof use the hypothesis?

--- Paul R. Halmos

Memory Quizzes:      Once a week there will be a short quiz over the newly defined terms and simple proofs from the book. The best way to memorize these is to actually understand them. Mathematics is not a series of incantations.

Relevancy Presentations:      Students should pick a topic that illustrates the intersection of this course with their interests in Math, MathEd, or some other field. Some example topics are below. Remember to give a good talk as you should have learned in CMST191/192 or MATH/STAT/DSCI395. If you don’t know how, ask the instructor.
Important Dates: Project Topic Okayed by March 4, Best Version of Slides Due April 15, Presentation Given May 1.

·         Algebraic Coding and Reed-Solomon Codes

·         Impossibility of Angle Trisection, Squaring the Circle, and Doubling the Cube

·         Insolvability of the Quintic by Radicals

·         Constructible Regular n-gons

·         Matrix Groups in Quantum Mechanics

·         Frieze Groups, Cystallographic Groups, and Plane Periodic Patterns

·         Cayley Digraphs of Groups with Applications

·         Classification of Finite Simple Groups and the Monster Group

Exams:     There will be in-class exams at the completion of each chapter. Exam dates are tentative until officially announced in class. The final exam will consist of a comprehensive in-class final exam and a take-home exam. The in-class portion of the final exam is scheduled for Monday, May 6, 8:00am – 10:00am. The take-home portion of the final exam will have a due date given by the instructor near the end of the semester.

Brightspace:   Many course materials can be found on Brightspace including homework problems, study materials, and approximate grades. If at any point during the semester you would like to know your exact grade, please email the instructor.

Late/Missed Work: Late textbook activities, homework or missed exams will result in a score of zero. Make-up exams before the time of the normal exam will be given at the discretion of the instructor. If you miss class, it is your responsibility to obtain notes and assignments from fellow students. If you have an unavoidable absence, please inform the instructor beforehand.

Study Groups:           Students are allowed to form study groups for the course. However, students are strongly encouraged to work on the homework individually first. All students must put homework solutions into their own words. Copy and pasted typed homework (even with minor changes) will be considered cheating. Do your own written homework.

Academic Dishonesty:  Any type of academic dishonesty (cheating, copying, discussing confidential oral-exam problems with other students, using a solutions manual to do homework, finding solutions online, etc.) will result in failure and will be reported to school authorities. If you are having trouble with an assignment, please see the instructor first.

Note:   This syllabus is subject to change if deemed necessary by the instructor.

 

Tentative Schedule of Events – Math 447

(as of April 17, 2019; subject to change)
Note: This is very tentative and will depend on our pace. Here is what is for sure:
1) There will be exams, usually at the end of chapters.
2) The Relevancy Presentation deadlines and Final exam time are fixed.

 

Week Beginning

Monday

Wednesday (1st Hour)

Wednesday (2nd Hour)

Friday

1/14

Introductions

Number Theory Review

Number Theory Review, cont

 

1.11, 24, 25 (2,4), 32, 39

2.1
 Rings

Finish the Activity

2.1, cont.

1/21

NO CLASS
MLK Day

Oral Homework

 

Written:

2.1, cont.

2.15 (2), 16 (6), 25, 114 (1, 2, 3, 4, 5, 7, 8, 9, 10), 115,

Memory Quiz &
2.2
Subrings

2.34(1), 2.37(3), 2.124

1/28

2.3
Integral Domains and Fields

 

Euclidean Algorithm in Z and Q[x]

 

2.42, 2.121
Inverse of: 225 mod 1393, 40108 mod 165973
Find r(x) and s(x) so that:
r(x) (2x4+3x3-5x2-6x-1)
+s(x)(2x3+x2-6x-2)=1

FREEZE OUT

FREEZE OUT

Oral Homework

 

Written:

2/4

Memory Quiz 
2.4
Ideals

 

Find 7x5-4x3+6x2-8 mod x2+7x+1 in Reals[x],
2.74(1), 2.75(1), 2.77(1,2,3,4), 2.122(1,2), 2.125(1), 2.130, 2.134

2.4, cont.

2.5
Quotient Rings

 

2.135 (3), 2.136

Finite Fields Activity

 

Finish the ones we didn’t get to.

2/11

Oral Homework

 

Written: 2.122 (1), 2.136 (1,2)

Memory Quiz

2.6
Ring Homomorphisms and Isomorphisms

2.6, cont

2.6, cont

 

Isomorphism Activity

 

2.96(1,2), 2.101(1), 2.108(1.1, 1.2), 2.109(1,2,3), 2.127, 2.137

2/18

Oral Homework

 

Written:

Memory Quiz

Chapter 3: Vector Spaces

Vector Spaces, cont.

Chapter 2 Exam

2/25

BLIZZARD

Go over Exam

 

Groups you should (already) know

Subgroups of Sn activity

 

HW: Judson Text Chapter 5 Exercises
#2dgjo, 10, 13, 23, 30ab, 31

Group Properties

 

Subgroups
(Judson Chapter 3)

 

HW: Judson Chapter 3 Exercises
#15, 16, 25, 31, 33, 39, 42, 48, 53, 54

3/4

Relevancy Projects Topics Okayed

 

Oral Homework

 

Written:


Memory Quiz

Cyclic Groups
(Judson Chapter 4)

 

#1ce, 2, 3f(Hasse Diagram Only), 4abc, 9, 11, 14, 23, 26, 29, 31, 37, 39

Intro to Elliptic Curves

3/11

SPRING BREAK

3/18

Properties of Sn

HW: Judson Text Chapter 5 Exercises
#3, 11, 18, 25, 34, 35

Oral Homework

 

Written:

4.2
Cosets, LaGrange’s Theorem
(Judson Chapter 6)

Memory Quiz

4.2, cont.

 

HW: Judson Text Chapter 6 Exercises
#5abeh, 6, 9, 11, 12, 17, 18, 20

3/25

Group Isomorphisms
(Judson Chapter 9)

HW: 2, 9, 11, 14, 23, 35, 36, 38, 45

Oral Homework
(Just from Chap 6)

Memory Quiz

Review

Chap 9, cont.

EXAM 2
(Judson Chaps 3 -6)

4/1

Chap 9: Direct Products

HW: 16, 18, 20, 22, 24, 25, 47, 48

Go Over Exam

Fundamental Theorem of Finite Abelian Groups Activity

Oral Homework

No Memory Quiz

4/8

Chap 10: Normal Subgroups and Factor Groups

HW: 1ace, 4, 7, 9, 11, 13cd, 14

Chap 11: Homomorphisms

HW: 2, 4, 5, 7, 13, 16, 18, 19

Snake Lemma Activity

Oral Homework

4/15

Relevancy Slides Due

 

Oral HW, cont.

 

Memory Quiz (Chaps 9, 10, 11, 13.1)

 

Chap 11:

Isomorphism Theorems

 

HW: 17

Elliptic Curves
Intro and Basics

 

HW: Problems 1 - 4

NO CLASS

Spring Break Day

4/22

Number Theory Problems associated to Elliptic Curves

 

HW: Problems 5 - 8

Rational Elliptic Curve Activity

Elliptic Curves over Finite Fields

Finite Field Elliptic Curve Activity

4/29

Homework Day

Relevancy Presentations

Relevancy Presentations

Final Review
Take Home portion handed out


Final Exam (In-Class Portion)

Monday, May 6

8:00am – 10:00am

 

 

 

Commitment to Inclusive Excellence

WSU recognizes that our individual differences can deepen our understanding of one another and the world around us, rather than divide us. In this class, people of all ethnicities, genders and gender identities, religions, ages, sexual orientations, disabilities, socioeconomic backgrounds, regions, and nationalities are strongly encouraged to share their rich array of perspectives and experiences.  If you feel your differences may in some way isolate you from WSU’s community or if you have a need for any specific accommodations, please speak with the instructor early in the semester about your concerns and what we can do together to help you become an active and engaged member of our class and community.

If you or a friend has been a victim of sexual assault, dating violence, domestic violence, or stalking, you can talk to a trained, confidential advocate by calling 507-457-5610. See the Sexual Violence page for more information about your rights and resources.

Campus Resources

 

The Standard Disclaimer applies.

© Eric Errthum, April 2019, all rights reserved.