MATH 0430Introduction to Abstract Algebraic Systems2104 RevisedINSTRUCTORDr. Elayne ArringtonOffice: 608 Thackeray HallOffice Hours: MWF 3:00-5:00 pm (or by appointment )Telephone: 412-624-8337 or 8375 (Math Office)fax: 412-624-8397e-mail: [email protected] SCHEDULEThis course meets MWF 2:00-2:50 p.m.COURSE WEBSITE http://www.math.pitt.edu/~earrCOURSE DESCRIPTIONThis course introduces the student to abstract algebraic concepts: rings, integral domains, fields, integers, rational numbers, real and complex numbers and polynomials. Many examples will be presented during class and in the homework. The students are expected to improve their proof-writing techniques.COURSE GOALSThis course will familiarize the students with several basic algebraic structures.The students will be able form new structures from these basic structures.The course will provide students with the opportunity to develop and practice communication skills and work in groups or teams while using course concepts.The course will provide student with the opportunity to improve communication skills by writing solutions to many problems in the form commonly accepted in mathematical journals.The course will provide students with the opportunity to improve presentation skills by participating in a special project.TEXTBOOKThe textbook for Math 0430 is Abstract Algebra; An Introduction, second editionby Thomas W. Hungerford. We will cover most topics in chapters 1 through 6. It is important that the student reads the section(s) to be covered before the appropriate lecture.Prerequisites: The material in Appendices A through D.ATTENDANCEAlthough class attendance is not specified as a factor in determining the course grade, the student is expected to attend all classes and he or she is responsible for all assignments and material covered in class, and all announcements or changes made in class, whether present or not.HOMEWORKHomework is an essential part of the course and will be regularly assigned and graded. It is important that the student improve her/his proof-writing technique through the homework. Fifty (50) problems worth ten (10) points each will be assigned during the session. The solutions to the problems are to be written in the forms of proofs, as presented in the textbook and in class. Each assignment is due at the beginning of the appropriate class. Solutions will be provided when the homework is collected and no late homework will be accepted. The highest 40scores will be counted for the homework grade.EXAMINATIONSThree 50-minute examinations will be given.SPECIAL PROJECTEach student will participate in a special team-project that uses concepts learned early in the course. Working in groups of three or four, students will complete the project, submit a written analysis of the project and give an 8-10 minute presentation of the project in class.ArringtonMath04302104FINAL EXAMINATIONThere will be a comprehensive final examination.COURSE GRADEThe course grade will be based solely on the student’s performance on the homework, the examinations, the special project, and the final examination. The homework will comprise 25% of the grade, the three 50-minute examinations will count for 40% of the grade, the special project will count for 10% of the grade, andthe final examination will count for 25% of the course grade.STUDENTS WITH DISABILITIESA student with a disability for which he or she is requesting an accommodation, is encouraged to contact both the instructor and the Office of Disability Resources and Services, 216 William Pitt Union (412) 648-7890 as early in the term as possible.ACADEMIC INTEGRITYCheating/plagiarism will not be tolerated. Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction. Students may work together or use library resources to do homework, but each student must write his or her own solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly. DEADLINESThe student should be aware of all CAS deadlines. Add/drop period ends: Tuesday, January 19, 2010. Monitored withdrawal ends: Friday, March 5, 2010.ArringtonMath04302104MATH 0430Introduction to Abstract Algebraic SystemsCourse Schedule (2104) RevisedDate#Lecture Topic Text Reading DueW, Jan. 6 1 The Division Algorithm 1.1F, Jan. 8 2 Divisibility 1.2M, Jan.11 3 Divisibility 1.2W, Jan. 13 4 Primes and Unique Factorization 1.3 Homework #1F, Jan. 15 5 Primality Testing. 1.4M, Jan. 18 HolidayW, Jan. 20 6 Congruence and Congruence Classes 2.1 Homework #2F, Jan. 22 7 Modular Arithmetic 2.2M, Jan. 25 8The Structure of Ζp When p is prime2.3 SP(a) W, Jan. 27 Review Homework #3F, Jan. 29 Exam #1M, Feb. 1 9 Definition and Examples of Rings 3.1W, Feb. 3 10 3.1F, Feb. 5 11 Homework #4M, Feb. 8W. Feb. 10F, Feb. 12 12 Basic Properties of Rings 3.2M, Feb. 15 13 3.2W, Feb. 17F, Feb. 19 14 Isomorphisms and Homomorphisms 3.3M, Feb. 22 15 3.3 SP(b)W, Feb. 24 16 Polynomial Arithmetic and the Division Algorithm4.1 SP(c)F, Feb. 26 17 Divisibility in F[X] 4.2 Homework #5,M, Mar. 1 18 Irreducibles and Unique Factorization 4.3 Homework #6, SP(d)W, Mar. 3 19F, Mar. 5 20 Polynomial Functions, Roots, and Reduciblity4.4 SP(e)M, Mar. 15 Homework #7, SP(f), SP(g)W, Mar. 17 SP(g) F, Mar. 19 SP(g)M, Mar. 22Review for Exam #2W, Mar. 24Exam #2ArringtonMath04302104F, Mar. 26 21 Congruence in F[X] and Congruence Classes5.1M, Mar. 29 22 Congruence-Class Arithmetic 5.2W, Mar, 31 23F, Apr. 2 24 The Structure of F[X]/(p(x)) When p(x) is Irreducible5.3M, Apr. 5 25 5.3 Homework #8W, Apr. 7 26 Ideals and Congruence 6.1F, Apr. 9 27 Quotient Rings and Homomorphisms 6.2M, Apr.12 28 6.2 Homework #9W, Apr. 14 29 The Structure of R/I When I Is Prime or Maximal6.3F, Apr. 16 30 6.3M, Apr. 19Review for Exam #3Homework #10W, Apr. 21Exam #3F, Apr. 23Review for Final ExamFinal Examination: Saturday May 1, 2:00 – 3:50 p.m.ArringtonMath04302104Special ProjectWorking in pairs or groups of 3, students will use concepts of section 2.3 to develop a public-key system. This will be accomplished