MATH 153 -- ABSTRACT ALGEBRA -- FALL 2006:
Barus - Holley 153, 12:00 - 12:50pm (E block)
Professor Steven Miller (sjmiller AT math.brown.edu), Kassar
House, Room 210, 863-1123
FINAL EXAM: BH 141, 2 - 5PM, FRIDAY, DECEMBER 15^{TH}.
HOMEWORK:
Week One (9/6- 9/8):
Read:
Skim Sections 1.1, 1.3 of Mod p Arithmetic, Group Theory and
Cryptography, review the Induction and
Types of Proofs handouts, read Sections 1.3 and 1.4 of Lang.
HW: Due Monday 9/11:
Mod p Arithmetic Handout: Page 6: #1.2.16, #1.2.17; Lang: Page 5: #2 (hint:
#1 is useful: you may use this problem without proving it); Lang: Page 9: #1;
Lang: Page 14: #4,#5,#9 (hint: #8 is useful: you may use this problem without
proving it).
Week Two (9/10 - 9/15): Wednesday's Lecture Notes: click
HERE.
Read:
Skim Sections 1.4, 1.5 of Mod p Arithmetic, Group Theory and
Cryptography, Chapter 1 of Lang's book, and sections 2.1 and 2.2 of Lang.
HW: Due Monday 9/18:
Mod p Arithmetic Handout: Page 12: #1.4.8 (this was originally assigned for
Week One: if you already did it you do not need to do it again, but if you
couldn't do it you have a second chance), Page 13: #1.4.17; Page 14: #1.4.26; Lang: Page 24: #8
(there is a typo: the entry yx^{2} should be x^{2}y), #11, #14.
Week Three (9/18 - 9/22):
Read:
Lang: Chapter 2, Sections 2.1, 2.2, 2.3, 2.4
HW: Due Monday 9/25: Lang, Page 32:
#1, #4; Page 39: #4, #5, #11, #15 and prove that in general a translation is not
an automorphism.
Week Four (9/25 - 9/29:
Read:
Lang: Chapter 2, Sections 2.4, 2.5, 2.6.
HW: Due Monday 10/2: Lang: Page 51:
#4, #10, #18, #26, #28. If you will not be in class on Monday because of Yom
Kippur, you may hand in the HW on Wednesday.
Week Five (10/2 - 10/6):
Read:
Lang: Chapter 2, Sections 2.6, 2.7. Read carefully Theorem 7.1 (similar
to HW problem #15 from page 39 of Lang) and Lemma 7.3; these two results will be
used in class without proof.
HW: Due Wednesday 10/11: Lang, Page
57: #4, #10; Find all generators for (Z/pZ)* for p = 5, 7 and 11; Lang, Page 66:
#1af, #3, #5, #8; Lang, Page 70: #2, #3.
Week Six (10/11 - 10/13): No class 10/9 because of Columbus
Day:
Read:
Lang: Chapter 2, Sections 2.8, 2.9
HW: Due Monday 10/23: Lang, Page
78: #1, #2, #3, #4.
Week Seven (10/16 - 10/20): MIDTERM
WILL BE IN CLASS ON FRIDAY, OCTOBER 20^{TH}.
Read:
Lang: Chapter 2, Section 2.9 (especially the proofs of the first two Sylow
Theorems); Chapter 3, Sections 3.1.
HW: Due Monday 10/23: Lang, Page 82: #1,
#2, #3, #4a, #8.
Week Eight (10/23 - 10/27):
Read:
Lang: Chapter 3, Sections 3.1, 3.2, 3.3.
HW: Due Wednesday 11/1: Lang, Page 86: #1,
#3, #10; Lang, Page 89: #2, #4, #7; Page 97: #2, #6, #7, #18, #19.
Week Nine (10/30 - 11/3):
Read:
Lang: Chapter 3, Sections 3.3, 3.4; Chapter 4, Sections 4.1, 4.2.
We will probably go over Section 3.4 very quickly, so make sure you have
carefully read it before class.
HW: Due Wednesday 11/8: Lang: Page 104:
#1; Lang, Page 117: #1a, #8 (note that if p=2 then -1 = 1 in this field), #12;
Lang, Page 120: #3.
Week Ten (11/6 - 11/10):
Read:
Lang: Chapter 4, Sections 4.2, 4.3, 4.5, 4.6. We are skipping section 4.4,
though you should skim it (it will not be on any exam).
HW: Due Wednesday 11/15: Lang: Page 124:
#2, #4, #8, #16, #19; Lang, Page 141: #4, #6
Week Eleven (11/13 - 11/17):
Read:
Lang: Chapter 4, Sections 4.6, 4.7; Second Midterm will
be Fri, Nov 17th in class. It will only count if it is better than previous
midterm; however, all students are required to take the exam.
HW: Due Monday 11/21: Lang: Page 150: #1,
#2, #4, #10.
Week Twelve (11/20):
NO CLASS ON 11/22 BECAUSE OF THANKSGIVING RECESS.
Read:
Lang: Chapter 4, Sections 4.8.
HW: Review notes and look at some of the
suggested problems; you do not need to turn anything in, just look at some of
the problems. A good way to review much of the material we've done to date is to
study the Handout on Mersenne Primes, which we will cover next week.
Week Thirteen (11/27 - 12/1):
Read:
Lang: Chapter 4, Section 4.8, Handout on Mersenne Primes.
HW: Due Wednesday 12/6 (though if you hand it
in by 12/4, better chance of getting it back in time to study for the final):
(1) Rigorously prove that (t_{1}, t_{2}) is not a principal
ideal in F[t_{1}, t_{2}] when F is an arbitary field; Lang: PAge
164: #1.
Week Fourteen (12/4 - 12/8):
OPTIONAL PASS/FAIL SUPPLEMENTAL FINAL: IN CLASS ON FRIDAY.
Read:
Handout on Mersenne Primes
and Handout on Primes in Arithmetic Progressions.
HW: Study for the final exam on the 15th at 2pm.
HANDOUTS:
Mod p Arithmetic, Group Theory and Cryptography (from the first chapter of An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash, Princeton University Press 2006).
Notes on Induction, Calculus, Convergence, the Pigeon Hole Principle and Lengths of Sets (from the first appendix to An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash, Princeton University Press 2006).
Handout on Types of Proofs (from a handout I wrote for math review sessions at Princeton, 1996-1997; this was written for students from calculus to linear algebra).
Handout on Mersenne Primes, by myself and Sarah Meiklejohn: shows how many of our results about groups and rings can be used to analyze for what p is Mp = 2p - 1 a prime (such numbers are called Mersenne primes, and have many wonderful properties; for instance, all even perfect numbers are simply related to Mersenne primes).
Primes in Arithmetic Progressions (from chapter three of An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash, Princeton University Press 2006).
GENERAL REMARKS
DESCRIPTION: An introduction to the principles and concepts of modern abstract algebra. Topics will include groups, rings, and fields, with applications to number theory, the theory of equations, and geometry. MATH 153 is required of all students concentrating in mathematics.
GENERAL: Homework is to be submitted on time, neatly written and stapled; late, messy or unstapled homework will not be graded. Working in groups is encouraged; everyone must submit their own assignment and write the names of all group members on the first page. Algebra and group theory occur throughout mathematics; there is a lot of language and notation to learn, as well as results and applications. We shall explore many of the connections with results in cryptography and number theory; further applications (why there are no generalizations of the quadratic formula for polynomials of degree greater than 4, why you cannot trisect an arbitrary angle, double a cube or construct a regular 7-gon with just straight edge and compass) shall be covered in MATH 154. You should be spending at least one hour a day every day on this class.
GRADING: There will be one or two midterms and a final. Grading: approximately 20% homework, 30-40% midterms and quizzes, 40-50% final (2pm on December 15th), with bonus points for class participation.
TEXTBOOK: Serge Lang's Undergraduate Algebra, 3rd Edition, as well as supplemental handouts. We will cover chapter one of my book (An Invitation to Modern Number Theory) and the first four chapters of Lang's Undergraduate Algebra.
READING: Before each class you must read the relevant sections of the book or the handouts. This does not mean you must master the material before class, but rather that you should be familiar with the notation and the statement of the main results.
OFFICE HOURS: After class M-W-F and by appointment.
(EXTRA CREDIT) CHALLENGING PROBLEMS: I will give several extra credit problems throughout the semter; here are the first two: (1) given any positive integer x prove there is a positive integer y such that xy has only 0s and 1s for digits; (2) explain why http://digicc.com/fido/ correctly guesses your number. For extra credit, submit the solution(s) by November 1st. If you work in a group remember to mention your colleagues.
You should spend at least an hour a day every day on this course.
This is the syllabus from when I taught the course in Fall '05. The course in Fall 2006 will be similar, though you should check the webpage / class for changes. We will cover Chapter 1 of my book (An Invitation to Modern Number Theory) and Chapters 1 - 4 of Lang's Undergraduate Algebra.