Groups and Rings III

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Description

The algebraic notions of groups and rings are of great interest in their own right, but knowledge and understanding of them is of benefit well beyond the realms of pure algebra. Areas of application include, for example, advanced number theory; cryptography; coding theory; differential, finite and algebraic geometry; algebraic topology; representation theory and harmonic analysis including Fourier series. The theory also has many practical applications including, for example, to the structure of molecules, crystallography and elementary particle physics.

Objective

To introduce students to the basic ideas and methods of modern algebra. At the end of this course students should be able to understand the idea of a group, a ring and an integral domain, and be aware of examples of these structures in mathematics; appreciate and be able to prove the basic results of group theory and ring theory; understand and be able to apply the fundamental theorem of finite abelian groups; understand Sylow's theorems and be able to apply them to prove elementary results about finite groups; appreciate the significance of unique factorization in rings and integral domains.

Content

Topics covered are: (1) Groups, subgroups, cosets and normal subgroups, homomorphisms and factor groups, products of groups, finitely generated abelian groups, groups acting on sets and the Sylow theorems. (2) Rings, integral domains and fields, polynomials, ideals, factorization in integral domains and unique factorization domains.

Stuart Johnson Lecturer for this course

Delivery

36 hours of lectures and tutorials

Assessment

Ongoing assessment 30%, exam 70%.

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)
• Communication skills (6)

Linkage past

Prerequisite is MATHS 1007A/B Mathematics I (Pass Div I) or both MATHS 1007A/B Mathematics I (Pass Div II) and MATHS 2004 Mathematics IIM (Pass Div I). PURE MTH 2002 Algebra II is assumed knowledge but the necessary material is revised at the start of the course.

Linkage present

No present linkages have been noted.

Linkage future

This course is not a prerequisite for any specific course, but is a recommended course for students intending to take Honours Pure Mathematics. This course also provides a useful background for PURE MTH 3012 Fields and Geometry III .

Restrictions

None.

Recommended text

None.