A group consists of a set together with a binary operation which satisfies certain axioms, and a ring or field consists of a set together with two binary operations which satisfy certain axioms. Many important classes of mathematical objects can be regarded as groups, rings or fields. For example, permutations or symmetry transformations together with the operation of composition form groups, the integers together with the operations of addition and multiplication form a ring, and the rational, real or complex numbers together with the operations of addition and multiplication form fields.

In this module, various general definitions and theorems in group, ring and field theory will be studied, and then illustrated using specific examples.  Students will thereby be exposed to some of the fundamental structures and concepts of abstract algebra.

Recommended Modules: MA0212 Linear Algebra

• State the group axioms.
• Understand the concepts of abelian groups, subgroups, group homomorphisms, group isomorphisms, cosets, normal subgroups and quotient groups.
• Prove basic group theorems, such as Lagrange’s theorem and the first group isomorphism theorem.
• Work with certain examples of groups, such as cyclic, dihedral and symmetric groups.
• State the ring and field axioms.
• Understand the concepts of commutative rings, rings with an identity, integral domains, ring homomorphisms, ring isomorphisms, subrings, subfields, ideals, quotient rings and factorisation in commutative rings.
• Prove basic ring and field theorems, such as the first ring isomorphism theorem.
• Work with certain classes of rings and fields, such as polynomial rings, matrix rings and fields of fractions.

27 – 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Appreciation of the general prevalence and significance of groups, rings and fields in mathematics.

Ability to understand and apply the basic concepts of group, ring and field theory.

Transferable Skills:
Recognition of the power of mathematical abstraction and the importance of mathematical rigour.

Critical thinking and problem solving skills. 

Ability to present work in a scholarly manner.

Formative assessment is carried out by means of regular homework exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

• Groups, abelian groups, subgroups, group homomorphisms, group isomorphisms, cosets, normal subgroups and quotient groups.
• Basic group theorems, including Lagrange’s theorem and the first group isomorphism theorem.
• Examples of groups, including cyclic, dihedral and symmetric groups.
• Rings, commutative rings, rings with an identity, integral domains, fields, ring homomorphisms, ring isomorphisms, subrings, subfields, ideals, quotient rings and factorisation in commutative rings.
• Basic ring and field theorems, including the first ring isomorphism theorem.
• Classes of rings and fields, including polynomial rings, matrix rings and fields of fractions.

D. S. Dummit and R. M. Foote, Abstract Algebra (Third Edition), Wiley, 2004.