# MTH416 Algebraic Structures

Lead Faculty: Dr. Igor Ya Subbotin

## Course Description

A look at groups, rings and fields, as well as applications of these systems. Discusses equivalence relations, Lagrange's Theorem, homomorphisms, isomorphisms, Cayley's Theorem and quaternions. Also examines error correcting codes and issues of cryptography. Graphing calculator may be required.

## Learning Outcomes

• 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 rings and their analogy to the study of homomorphisms.
• 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.
• Describe some applications of abstract algebra: such as in avoiding problems of round off, in error correcting codes, for classifications of crystals, underlying issues behind public key cryptography.