A look at groups, rings and fields, as well as applications of these structures. Discusses equivalence relations, Lagrange;s Theorem, homomorphisms, isomorphisms, Cayley’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 Lagrange’s theorem.
- Discuss homomorphisms, or relationships between groups such as isomorphism and factor groups and Cayley’s theorem.
- Compare rings, integral domains, and fields; structures with two binary operations defined on them. Discuss Fermat’s and Euler’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.
- 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).
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