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).
Why Choose National University?
We’re proud to be a veteran-founded, San Diego-based nonprofit. Since 1971, our mission has been to provide accessible, achievable higher education to adult learners. Today, we educate students from across the U.S. and around the globe, with over 200,000 alumni worldwide.
Focus on one subject at a time — one month at a time — and finish your degree faster.
75+ Degree Programs
Choose from associate, bachelor’s, and master’s degrees, plus credentials and certificates.
On Campus or Online
Study when and where it’s convenient for you with evening, weekend, and 100% online classes.
Apply or transfer any time. Classes start monthly, and applications are accepted year round.
Attend class and learn onsite at one of over 20 locations in California.
As a Yellow Ribbon school, we offer tuition discounts to servicemembers and dependents.
School of Arts, Letters, and Sciences
“Our faculty are established scholars and creators in their own fields. They are dedicated to making your entry into their disciplines a successful one, preparing you for your career as well as graduate study.”School of Arts, Letters, and Sciences