INTRODUCTION TO ALGEBRAIC STRUCTURES
Prerequisite and degree relevance: Either consent of Mathematics Advisor, or two of M341, 328K, 325K (Philosophy 313K may be substituted for M325K), with a grade of at least C-. This course is designed to provide additional exposure to abstract rigorous mathematics on an introductory level. Students who demonstrate superior performance in M311 or M341 should take M373K RATHER THAN 343K Those students whose performance in M341 is average should take M343K before taking M373K Credit for M343K can NOT be earned after a student has received credit for M373K with a grade of at least C-.
Course description: Elementary properties of the integers, groups, rings, and fields are studied.
The number of topics should be kept modest to allow adequate time to concentrate on developing the students' theorem-proving skills. Some instructors will prefer to introduce groups before rings and some will reverse the order. In any case, below are some reasonable choices of topics. One should not try to cover all of these topics. It is very important to avoid superficial coverage of too many topics. All potential graduate students will take M373K, where it is possible to expect more and to do more.
Topics: Groups: Axioms, basic properties, examples, symmetry, cosets, Lagrange's Theorem, isomorphism. Homomorphisms, quotient groups, and the Fundamental Homomorphism Theorem.
Optional: Rings: Axioms, basic properties, examples, integral domains, and fields. rings and properties of fields.
Optional: More about polynomial
Other options: Groups acting on sets, characterization of the familiar number systems in terms of ring and field properties, and other applications of groups.
Durbin July 2000