MTH416 Algebraic Structures
Lead Faculty: Dr. Igor Ya Subbotin
Course DescriptionA look at groups, rings and fields, as well as applications of these structures. Discusses equivalence relations, Lagrangeandapos;s Theorem, homomorphisms, isomorphisms, Cayleyandapos;s Theorem and quaternions. Graphing calculator may be required.
- Discuss sets, subsets, and partitions and equivalence relations.
- Discuss examples of groups, properties of groups; operations on groups. The nature of orbits, cycles, the alternating group, cyclic groups, abelian groups, cosets and Lagrangeandapos;s theorem.
- Discuss homomorphisms, or relationships between groups such as isomorphism and factor groups and Cayleyandapos;s theorem.
- Compare rings, integral domains, and fields; structures with two binary operations defined on them. Discuss Fermatandapos;s and Eulerandapos;s theorems.
- Show how homomorphisms are involved in solving a polynomial equation.
- Discuss irreducible polynomials and their importance.
- Explain what is meant by quaternions and the historical significance of their development.
- Discuss the idea of homomorphism.
- Discuss the concepts of an extension field, and of algebraic elements and of transcendental elements and how these tie together to show that every non-constant polynomial has a zero in some field.
- andlt;span style=andquot;font-size:11.0pt;font-family:andapos;Times New Romanandapos;,andapos;serifandapos;;andquot;andgt;Demonstrate knowledge of why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings)