Introduction to Modern Algebra

University credit, 12 assignments, 2 exams, $684 tuition, $75 administration fee, Prerequisites: see below. This is course U600-441.

 

An introductory course in modern, or abstract, algebra. Approaches the central algebraic notions of rings and fields by starting with concrete examples of number systems, and develops the general structures with these examples in mind. Groups are introduced in the context of rigid transformations of simple geometric objects.

Prerequisites: Two courses beyond calculus, including a course in Linear Algebra. This course requires a level of mathematical maturity and precision of thought typically developed in courses beyond the level of calculus (or in Honors Calculus courses). Students who have taken proof-intensive courses such as Geometry or Advanced Calculus will find that preparation helpful, but this course is also appropriate as a first proof-oriented course. Before registering, please use this link to contact Director of Mathematics.

Course Materials

• Numbers and Symmetry: An Introduction to Algebra, Johnston and Richman, 1997, CRC Press, ISBN: 084930301X. Available through bookstores.
• Hints for homework problems (always under construction)
• Student Handbook, available at learn.wisconsin.edu/il/studenthandbook/studenthandbook.pdf, or on paper by request to Learner Services, toll-free 1-877-UW-LEARN (877-895-3276).
• Syllabus and additional materials will be sent by the instructor. The first three lesson assignments can be found here.
• You might be interested in Fred Richman's web site at www.math.fau.edu/Richman/

Format of Course

The course consists of 12 lessons and 2 exams. Each lesson consists of:

• A reading assignment from the text.
• Recommended problems from the text, to test your understanding.
• A written assignment from the text, to submit for grading and feedback.

Exams are mailed to a proctor selected by the student in accordance with Independent Learning's guidelines. Students may request a re-take of either exam within two weeks of receiving their graded exam back. If an exam is retaken, the higher of the two grades will be the grade of record.

Grading

Score	Letter Grade
NOTE: The grading scale was revised on Nov 13, 2003. This grading scale applies to all students who complete the course after that date.
93-100	A
90-92	AB
83-89	B
80-82	BC
73-79	C
65-72	D
0-64	F

Each lesson and each exam is graded on a 100-point scale. The final score is computed as a weighted average, with the average of the lessons counting for 70%, and the average of the exams counting 30%. The exam average must be at least 40 for a passing grade. Assuming this is the case, the average of all your work is converted to a letter grade as shown in the table.

Course Outline

• Lesson 1: Mathematical Induction
• Lesson 2: Extensions of the Integers
• Lesson 3: Norms, Polynomials, Division Algorithm
• Lesson 4: The Euclidean Algorithm
• Lesson 5: Orders of Elements, Units in Rings
• Lesson 6: Primes, Unique Factorization
• Mid-Course Exam
• Lesson 7: Symmetries; Cyclic, Dihedral and Permutation Groups
• Lesson 8: Abstract Groups, Subgroups, Cosets, Isomorphism
• Lesson 9: Group of Units of a Finite Field, Euclidean Groups
• Lesson 10: Plane Lattices, Space Groups, Wallpaper Patterns
• Lesson 11: Fields and Field Extensions, Vector Spaces
• Lesson 12: Parity Checking, Linear Error-Correcting Codes
• Final Exam