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PIEDMONT COLLEGE - DEPT. OF MATHEMATICS Course Syllabus: Math 315.1 Modern Abstract Algebra (3hrs) Spring Semester 2003 MWF 9:00-9:50 (period 2) Stewart Hall 205 Instructor : William Kallfelz (Assistant Prof. of Math & Physics) Office Hours: Stewart Hall 229 (see enclosed copy of my schedule) . Phone/email: (706)778.3000 EXT 1310 Fax.: (706) 776 2811 •

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[email protected]

You may also try me at: [email protected] ONLY IF the @piedmont.edu is temporarily down. Otherwise, please correspond only at [email protected] My mailbox is in Daniel Hall, by the mail room.

I.) OBJECTIVES AND GOALS OF THIS COURSE: •

To gain an adequate introduction and survey of the unique methods of proof, calculation, and application involved in Modern Algebra, (appropriate to the level of rigour for an upper-division undergraduate course) as it interfaces with modern mathematics as a whole.

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To investigate the most important and frequently occurring algebraic structures (groups, rings, integral domains, fields, extension fields, etc.) in a variety of applications culminating in the theory of polynomial equations not solvable by radicals.

II.) COURSE INFORMATION AND POLICIES: •

Text: (required)

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Optional Text(s): Modern Algebra is a vast and beautiful subject. The following list are just a few samples of sources (no listed in any particular order) the interested student may care to refer to, depending on her or his particular interests:

John B. Fraleigh A First Course In Abstract Algebra (6th ed), Addison-Wesley .(2000) ISBN # 0 201 33596 4

Basic/Introductory (On the level of Fraleigh) Robert Redfield Abstract Algebra: A Concrete Introduction Addison-Wesley (2001) ISBN 0 201 43721 X (**Later in the course, when we study rings, I will refer to some problems and sections in this text**) Frank Ayers Modern Abstract Algebra (Schaum’s Outline) McGraw Hill (1986) ISBN 0 700 2655 6 W R Scott Group Theory Dover Publ (NY 1987) ISBN 0 486 65377 3 Advanced (The student may refer to these not so much as study, rather as reference and exposure into a few ‘classics’ in the field. Some of these texts, however, contain unusual and interesting expansions and elaborations of topics covered in class. Others present self-contained optional topics that may broaden the student’s interest)

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R Dean Classical Abstract Algebra Harper & Row (1990) ISBN 0 060 41601 7 Garret Birkhoff & McClaine Algebra Chelsea Publishing Co (1993 reprint) ISBN 0 828 40330 9 (** A classic**) Seth Warner Abstract Algebra (vol 1& 2 bound) Dover Publishing (1978) (** An excellent ‘Algebraists’ Desk Reference’ at a bargain price!**)

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Technology Aids: Abstract Algebra (thankfully) is one area of modern mathematics simple and elegant enough to be taught and learned using traditional methods and tools (bordwork, pencil, paper, etc.) However, interesting websites abound with nice presentations of certain of its applications, especially in (for instance) group theoretic methods applied to crystallography and plane isometries. URLs of these websites shall be announced in class at the appropriate time. Naturally, the student is encouraged to visit them.

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Prerequisite: Math 200 (Discrete Mathematics) and/or Math 213 (Calculus III ) or equivalent**

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Academic Integrity: From the Catalog (p. 40) “…the College imposes strict penalties for academic dishonesty, which is defined as follows • Cheating: Intentionally using or attempting to use unauthorized materials, information or study aids in any academic exercise. • Fabrication: Intention and unauthorized invention or falsification of any information or citation in an academic exercise. • Facilitating Academic Dishonesty: Intentionally or knowingly helping or attempting to help another to commit and act of academic dishonesty. • Plagiarism: Intentionally or knowingly representing the words or ideas of another as one’s own in any academic exercise. …Details on administrative procedures…may be found in the Piedmont College Student.”

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Attendance: Regular and punctual attendance is necessary and expected. Roll will be taken at least once per week. For maximum efficacy punctuality is essential as daily lessons plans include emphasis on in-class work. No student is allowed to remain in class after two consecutive weeks of class absence, and is therefore subject to mandatory withdrawal irrespective of drop date. Under unusual circumstances (illness, etc.) the student is expected to communicate with the instructor, and provided less than four consecutive classes were missed by the student, arrangements can be made at the consent of the instructor regarding making up missed work.

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Make-Up Policy: While a student may take exam prior to exam date, it is left to the discretion of the student to consult with instructor concerning extenuating circumstances preventing a student from taking exam at announced date(s). Should unusual circumstances prevent student from taking exam before or during exam date, it is left to the instructor to decide possible arrangements of student's grade, contingent on student's consultation with instructor.

The idea is that the student has gained a measure of “mathematical maturity” appropriate at an advanced sophomore level. AT THE VERY LEAST THIS MUST INCLUDE Math212 (Calculus II). Discrete mathematics provides a useful survey into some of Modern Algebra’s essential features (set theory, logic and proofs), but can be taken concurrently. A brief survey of these methods shall be covered in the introductory section of this course.

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III.) GRADING: A.) Three 1 hour exams (worth 100 pts) will be administered. (the lowest exam score will be dropped) B.) Four homework sets shall be assigned (@25 pts each) throughout the semester for a total of 100 points. C.) A paper and presentation shall be assigned (due at end of semester) worth 100 points D.) the final exam is worth 200 points F.) The student's final letter grade will be assigned indicated in the following table (based on a 12.5% scale):

Total Points (Pts) 525 < Pts < 600 450 < Pts < 525 375 < Pts < 450 300 < Pts. < 375 Below 300

FINAL LETTER GRADE A B C D F

IV.) IMPORTANT DATES: Exam I: Wed Feb 12 Exam II: Mon March 17 Exam III: Wednesday April 9 Last day to hand in project: Friday May 02 Last day to withdraw and receive a W grade: Tuesday, March 4 Final Exam: Tuesday May 6 8:00-10:00 AM Martin Luther King Jr. Holiday: Monday, Jan 20. Spring Break: Week of March 10th-14th.. Good Friday Holiday: Friday, Apr 18

V.) COURSE PLAN/CONTENT Project/Presentation: The student is to investigate a particular problem in greater depth and hand in a report due sometime during the last week of class. Details about the report format will be handed out along with a list of possible topics.. The student may choose her or his own topic, but s/he should consult with the instructor beforehand. In general, keep in mind that: • •

Reports should be brief and concise (5-10 pages) preferably prepared in a format that’s presentation-ready. The goal of the assignment is to investigate any topic area involving abstract algebra in greater depth. (These areas can range from pure number theory to applied physics). Though I am not obviously looking for original research (though if it happens, terrific!) the student should obvously clearly focus on a specific topic that can be summarized and presented in 20 minutes. In addition, depending on the nature of the topic s/he selected, independent calculations are encouraged on the part of the student (meaning, the student is welcome to try developing an example suggested but not carried out in the topic he or she is investigating)

Report should also contain: a.) Statement of the problem b.) Relevance to certain area of study (cite specific instance(s))

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c.) Calculation section (if the calculations are elaborate or lengthy, they can included in an Appendix section) d.) Conclusion(s) •

Detailed Lesson Plan The lesson plan will follow the timetable below. Optional sections are listed in parenthesis notation. The optional material may or may not be covered in class (depending on time) however I will assign homework from the optional sections as well

Date

Section(s) Covered: (Fraleigh)

0. Introduction (Logic, sets, proofs about sets, mathematical induction, general mappings [functions] defined on sets) 1/10 Fri Introduction, 0.1-0.2 1/13 Mon 0.2 through 0.3 1/15 Wed 0.3 through 0.4 1/17 Fri 0.4 + handouts I. Elementary Algebraic Structures (Elementary theory of Semigroups, and Groups) 1/22 Wed 1.1 1/24 Fri 1.2 (homework 1 due) 1/27 Mon 1.3 1/29 Wed 1.4 1/31 Fri 1.5 2/03 Mon 2.1 2/05 Wed 2.2 2/07 Fri 2.3 2/10 Mon 2.4 2/12 Wed Exam I: (chapters 0,1, 2.1-2.4) 2/14 Fri 3.1 (homework 2 due) 2/17 Mon 3.2 2/19 Wed 3.2 through 3.3 2/21 Fri 3.3 through 3.4 2/24 Mon 3.4 through 3.5 2/26 Wed 3.5 II. Topics In Advanced Group Theory 2/28 Fri 4.1 3/03 Mon 4.1 through 4.2 3/05 Wed 4.2 3/07 Fri 4.2 through 4.3 (selections) 3/17 Mon Exam II: (chapters 2.3-2.4, 3.1-3.5, 4.1-4.2)) 3/19 Wed 4.5 3/21 Fri 4.5 through 4.6 (homework 3 due) 3/24 Mon 4.6 III. Advanced Algebraic Structures (Rings, Integral Domains, Fields, Extension Fields, Galois Theory) 3/26 Wed 5.1 3/28 Fri 5.2 3/31 Mon 5.3 4/02 Wed 5.3,5.4 cont. + selections (2.5 Redfield) 4/04 Fri 5.4, 5.5 + selections 4/07 Mon further selections (Redfield, 2.5, 2.6) 4/09 Wed Exam III: (ch 5, + Redfield) 4/11 Fri 6.1 4/14 Mon 6.2 4/16 Wed 8.1

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4/21 Mon 4/23 Wed 4/25 Fri 4/28 Mon 4/30 Wed 5/02 Fri

8.2 (homework 4 due) 8.3 8.5 Intro Galois Theory ( ch 25,27 Redfield) Presentations and Galois (cont) Presentations and Galois /Course Review

VI.) A FINAL NOTE: • Note: The syllabus provides a general plan for the course, deviations may be necessary. The lectures summarize, outline, and elaborate on the fundamental principles covered in each section, which may include some critical deviation and/or elaboration in areas where I need further elaboration. All this is not meant you are required to painstakingly pore over every page in the text indiscriminantly. The student should aim for critical understanding quality Only by regular attendance will the student distinguish precisely what points are emphasized WELCOME AND GOOD LUCK! (“May our study be full of Light.” -Indian Proverb, ca. 2500 BC).