MATH 622 Selected Topics in Modern Algebra I

1. Course DescriptionSELECTED TOPICS IN MODERN ALGEBRA IMA 622This course extends the concepts that are taught in an undergraduate introduction to abstractalgebra. Topics include: finite and infinite groups, rings, ideals, and integral domains andfields.2. Goals of the Course1. To expand the undergraduate abstract algebra concepts2. To introduce additional topics such as rings and fields3. To prepare students for additional mathematics courses, possibly at the doctoral level3. Instructional Procedures1. Lecture/discussion2. Small group study3. Use of computer software4. Course ContentI. Review of Mappings and Operationsa. Mappingsb. Composition, invertible mappingsc. Operationsd. Composition as an operationII. Review of Groupsa. Definition and examplesb. Permutationsc. Subgroupsd. Groups and symmetryIII. Equivalence, Congruence, Divisibility'a. Equivalence relationsb. Congruence, the division algorithmc. Integer modulo nIV. Groupsa. Elementary propertiesb. Generatorsc. Cyclic Groupsd. Direct Productse Cosets

f. Normal Subgroupsg. LaGrange's TheoremV. Homomorphism and Factor Groupsa. Homomorphismb. Isomorphism and Cayley's Theoremc. Factor Groupsd. Factor-Group Computations and Simple Groupse. Groups in GeometryVI. Introduction to Ringsa. Definition and Examplesb. Integral domains, subringsc. Fieldsd. IsomorphismVII. The Familiar Number Systemsa. Ordered Integral Domainsb. The Integersc. Field of Quotientsd. Field of Rational Numberse. Ordered Fieldsf. Field of Real Numbersg. Complex Roots of Unity5. Evaluation Measures for Determining Students’ Grades1. Tests and Quizzes 50%2. Homework assignments and projects 15%3. Final exam 35%Note: Individual instructors may weigh evaluation measures differently.6. BibliographyA. Required TextFraleigh, John B. A First Course in Abstract Algebra, 5 th ed., Reading, MA, Addison-Wesley, 1994.Note: In mathematics courses it is usually preferable to have a designated textbookwhich helps to focus the discussion and standardize the language and symbolism.B. Additional Required ReadingsNoneC. Supporting BibliographyArtin, M. Algebra, Englewood Cliffs, NJ, Prentice-Hall, 1991.

Gallian, J.A. Contemporary Abstract Algebra 2d ed., Lexington, MA, D.C. Heath,1990.Hillman, A.P. and Alexanderson, G.L. Abstract Algebra: A First UndergraduateCourse, 5 th ed., Boston MA, PWS Publishing, 1994.McCoy, N.H. Rings and Ideals, Carus Monograph No. 8, The MathematicalAssociation of America, 1948.McCoy, N.H. and Janusz, GJ. Introduction to Modern Algebra, 5 th ed., Dubuque IA,Wm. C. Brown Publishers., 1992.D. Relevant Periodical SourcesNoneE. Relevant SoftwareExploring Small Groups