Introduction to Modern Algebra - Fall 2013

Arthur Cayley, 1821-1895

Niels Henrik Abel, 1802-1829

Evariste Galois, 1811-1832

Emmy Noether, 1882-1935


The fall 2013 Introduction to Modern Algebra class.

COURSE: MATH 5127-001, Call # 83784

TIME AND PLACE: 9:45-11:05 TR in Lamb Hall room 331

INSTRUCTOR: Dr. Robert Gardner OFFICE HOURS: 2:05-3:00 TR

OFFICE: Room 308F of Gilbreath Hall

PHONE: 439-6979 (308F Gilbreath), Math Department Office 439-4349

E-MAIL:gardnerr@etsu.edu
WEBPAGE: http://faculty.etsu.edu/gardnerr/gardner.htm (see my webpage for a copy of this course syllabus, copies of the classnotes in PDF, and updates for the course).

TEXT: A First Course in Abstract Algebra, 7th edition, by John B. Fraleigh (2002).

CLASS NOTES: We will use overheads for component of the lecture consisting of definitions, statements of theorems, and some examples. I will use the white board for additional examples and proofs of theorems. Copies of the notes are online at: http://faculty.etsu.edu/gardnerr/4127/notes.htm It is strongly recommended that you get printed copies of the overheads before the material is covered in class. This will save you from writing down most notes in class and you can concentrate on listening and supplementing the notes with comments which you find relevant. You should read the online notes to be covered in class before each class (we may not have class time to cover every little detail in the online notes). Try to understand the definitions, the examples, and the meanings of the theorems. After each class, you should read the section of the book covered in that class, paying particular attention to examples and proofs.

ADDITIONAL REFERENCES:
Groups and Their Graphs by Israel Grossman and Wilhelm Magnus, New York: Yale University Press, 1964. Our text does not give many details or examples of Cayley digraphs (in Section 7), but this book goes into a reasonable amount of detail and includes many more examples.
Visual Group Theory by Nathan Carter, New York: Mathematical Association of America, 2009. I will use this resource for some motivational and geometric examples.
Algebra by Thomas W. Hungerford, New York: Springer-Verlag, 1974. This is a standard graduate level algebra textbook. I will lightly rely on it for deeper results and proofs (some of which are omitted from out text).
A History of Abstract Algebra, by Isreal Kleiner, Boston: Birkhauser, 2007. As time permits, I will insert some historical comments and this is be a reliable source of such information.

PREREQUISITES: As the ETSU catalog states, the prerequisites for this class are Linear Algebra (MATH 2010) and Mathematical Reasoning (MATH 3000). We will depend heavily on both of these prerequisites, especially Mathematical Reasoning. Several examples will require knowledge of matrices. We will also have examples related to modular arithmetic and complex numbers, but this material will be reviewed in the class.

ACADEMIC MISCONDUCT: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Some of the homework problems are fairly standard for this class, and you may find proofs online or in an online version of the solutions manual. The online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class. If I get homework from two (or more) of you that is virtually identical, then neither of you will get any credit. If you copy homework solutions from an online source, then you will get no credit. These are examples of plagiarism and I will have to act on this as spelled out on ETSU's "Academic Integrity @ ETSU" webpage: http://www.etsu.edu/academicintegrity/faculty.aspx (last accessed 7/21/2013). To avoid this, do not copy homework and turn it in as your own!!! Even if you collaborate with someone, if you write the homework problems out in such a way that you understand all of the little steps and details, then it will be unique and your own work.

GRADING: Your grade will be determined by the average on three tests (T1-T3), and homework (HW). Your average is determined by

AVERAGE = (T1 + T2 + T3 + 3HW)/6.
Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate (based on grade points assigned by the university, on a plus/minus 3 point system). Each test will be 50% short answer/state a definition/give an example/computation and 50% proofs. Occasionally, the homework assignments will include bonus problems, but there are no other options for extra credit.

THE FINAL: We will use the time for the final to take a third (noncomprehensive) test on Thursday, December 12 from 8:00 a.m. to 10:00 a.m.

DESIRE2LEARN: I will not rely on the Desire2Learn ("elearn") website. Instead, I will simply post all material directly on the internet.

SYLLABUS ATTACHMENT: You can find an on-line version of the university's syllabus attachment (which contains general information concerning advisement, honor codes, dropping, etc.) at: http://www.etsu.edu/reg/documents/PDF/Syllabus_Attachment.pdf (last accessed 7/21/2013).

NOTE: This class is probably the most abstract math class you will take in your undergraduate curriculum! It covers material from the area of mathematics known both as "abstract algebra" and "modern algebra." Both terms are accurate; the second deserves some explanation. This class does not deal with "classical algebra" in the sense of dealing with polynomials, quadratic equations, and "solving for x." However, the topics of this class are inspired by these classical problems and ultimately our topics do deal with these topics (as well as many others). Primarily, this class deals with groups and rings. An additional topic is fields, and this area is dealt with mostly in the second course in the introductory algebra sequence (namely, Introduction to Modern Algebra 2, MATH 4137/5137). This class is a prerequisite for the graduate level Modern Algebra 1 (MATH 5410).

The rule of thumb is that you should allot an appropriate amount of time for your studies, at the freshman and sophomore level, at least 2 hours outside of class for each hour spent in class. This is a minimum for a class of this level!

IMPORTANT DATES (see http://www.etsu.edu/etsu/academicdates.aspx for the official ETSU calendar; last accessed 7/21/2013):
Monday, September 2 = Labor Day Holiday.
Friday, September 6 = Last day to drop without a grade of "W."
Tuesday, October 1 = Test 1 (0, I.1-I.7).
Monday and Tuesday, October 14 and 15 = Fall Break Holiday.
Monday, October 21 = Last day to drop without dean's permission.
Thursday, November 7 = Test 2 (II.8-II.11, III.13-III.15).
Thursday and Friday, November 28 and 29 = Thanksgiving Holiday.
Wednesday, December 4 = Last day to withdraw from the university.
Friday, December 6 = Last day of class.
Thursday, December 12 = Final (Test 3, IV.18-IV.23), 8:00 a.m. to 10:00 a.m. (the final exam schedule is online at http://www.etsu.edu/reg/registration/finalexam.aspx; last accessed 7/21/2013).

Other Supplemental Material

Our tentative schedule is as follows:

DAY
SECTION
TUE 8/27
Introduction;
Supplement: Why am I in this Class?
THR 8/29
0. Sets and Relations
TUE 9/3
I.1. Introduction and Examples
THR 9/5
I.2. Binary Operations
TUE 9/10
I.3. Isomorphic Binary Operations
THR 9/12
I.4. Groups
TUE 9/17
I.5. Subgroups
THR 9/19
I.6. Cyclic Groups
TUE 9/24
I.7. Generating Sets and Cayley Digraphs
THR 9/26
Review
TUE 10/1
Test 1 (0, I.1-I.7)
THR 10/3
II.8. Groups and Permutations
TUE 10/8
II.9. Orbits, Cycles, and the Alternating Groups
THR 10/10
II.9. cont.
TUE 10/15
Fall Break
THR 10/17
II.10. Cosets and the Theorem of Lagrange
TUE 10/22
II.11. Direct Products and Finitely Generated Abelian Groups;
Supplement: Small Groups
THR 10/24
III.13. Homomorphisms
TUE 10/29
III.13. Homomorphisms
THR 10/31
III.14. Factor Groups
TUE 11/5
III.15. Factor-Group Compositions and Simple Groups;
Supplement: The Alternating Groups An are Simple for n ≥ 5
THR 11/7
Supplement: Simple Groups;
Review
TUE 11/12
Test 2 (II.8-II.11, III.13-III.15)
THR 11/14
IV.18. Rings and Fields
TUE 11/19
IV.19. Integral Domains
THR 11/21
IV.21. The Field of Quotients of an Integral Domain
TUE 11/26
IV.22. Rings of Polynomials
THR 11/28
Thanksgiving Holiday
TUE 12/3
IV.23. Factorization of Polynomials over a Field
THR 12/5
Review
THR 12/12
Test 3 (IV.18-IV.23)

Homework

Section
Problems
Solutions
Due Date
Points
0. Sets and Relations
I.1. Introductions and Examples
0.14b, 0.17, 0.19, 0.31
1.21, 1.32, 1.36, 1.38
Solutions
Friday, 9/6
5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40
I.2. Binary Operations
I.3. Isomorphic Binary Structures
2.A, 2.23a, 2.23b, 2.37
3.7, 3.27, 3.33a
Solutions
Friday, 9/13
5 + 5 + 5 + 5 + 5 + 5 + 5 = 35
I.4. Groups
I.5. Subgroups
4.6, 4.19b, 4.31, 4.35, BONUS: 4.36
5.33, 5.41, 5.51, 5.54
Solutions
Friday, 9/20
5 + 5 + 5 + 5 + 5 +5 + + 5 + 5 + (5) = 40 + (5)
I.6. Cyclic Groups
I.7. Generating Sets and Cayley Digraphs
6.21, 6.46, 6.50
7.3, 7.9
Solutions
Friday, 9/27
5 + 5 + 5 + 5 + 5 = 25
II.8. Groups and Permutations
II.9. Orbits, Cycles, and the Alternating Groups
8.4, 8.20, 8.46
9.3, 9.11, 9.27a
Solutions
Friday, 10/11
5 + 5 + 5 + 5 + 5 + 5 + 5 = 30
II.10. Cosets and the Theorem of Lagrange
II.11. Direct Products and Finitely Generated Abelian Groups
10.6 & 10.7, 10.15, 10.35, 10.40
11.4, 11.26, 11.47, 11.51, Bonus: 11.18
Solutions
Friday, 10/25
5 + 5 + 5 + 5 + 5 + 5 + 5 + (5) = 40 + (5)
III.13. Homomorphisms
III.14. Factor Groups
13.6, 13.29, 13.37, 13.50, 13.52, Bonus: 13.49
Bonus: 14.2
Solutions
Friday, 10/25
5 + 5 + 5 + 5 + 5 + (5) + (5) = 25 + (10)
III.14. Factor Groups
III.15. Factor-Group Compositions and Simple Groups
14.10, 14.26, 14.31
15.2, 15.40
Solutions
Friday, 10/25
5 + 5 + 5 + 5 + 5 = 25
IV.18. Rings and Fields
IV.19. Integral Domains
18.12, 18.41, 18.48, 18.50, BONUS: 18.A
19.12, 19.23, 19.29
Solutions
Friday, 11/22
5 + 5 + 5 + 5 + 5 + 5 + 5 + (5) = 35 + (5)
IV.21. The Field of Quotients of an Integral Domain
IV.22. Ring of Polynomials
21.2, 21.9, 21.12(a), 21.12(b)
22.14, 22.26, 22.29, BONUS: 22.31(b)
Solutions
Friday, 11/22
5 + 5 + 5 + 5 + 5 + 5 + 5 + (5) = 35 + (5)
TOTAL
-
-
-
330 + (20)
The numbers in parentheses represent bonus problems.


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Last updated: December 3, 2013.