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

Some fundamental definitions and results in group theory were studied in the Year Two module MA0213 Groups.  This Year Three module will provide a reinforcement and extension of material from Year Two, together with the introduction and study of important definitions, theorems and examples for rings and fields.  Students will thereby be exposed to a variety of the basic structures and concepts of abstract algebra.

Recommended Modules: MA0212 Linear Algebra, MA0213 Groups

• State and apply the group, ring and field axioms.
• Fully understand basic definitions and theorems related to groups, including abelian groups, cyclic groups, dihedral groups, symmetric groups, subgroups, group homomorphisms, group isomorphisms, cosets, normal subgroups, quotient groups, Lagrange’s theorem, the first group isomorphism theorem and Cayley’s theorem.
• Understand the concepts of commutative rings, rings with an identity, zero divisors, integral domains, units, characteristics, finite fields, ring homomorphisms, ring isomorphisms, subrings, subfields, ideals, principal ideals, maximal ideals, quotient rings and the first ring isomorphism theorem.
• Understand and work with certain classes of rings and fields, including matrix rings, Euclidean domains, rings of polynomials and fields of fractions of integral domains.
• Understand the basic theory of field extensions, including degrees of extensions, finite extensions, simple extensions, algebraic elements, transcendental elements, algebraic extensions, adjoining subfields and splitting fields.

27 fifty-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.

Skills:
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. 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.

• Revision of basic definitions and results for groups, including abelian groups, cyclic groups, dihedral groups, symmetric groups, subgroups, group homomorphisms, group isomorphisms, cosets, normal subgroups, quotient groups, Lagrange’s theorem, the first group isomorphism theorem and Cayley’s theorem.
• Rings, commutative rings, rings with an identity, zero divisors, integral domains, units, characteristics, fields, finite fields, ring homomorphisms, ring isomorphisms, subrings, subfields, ideals, principal ideals, maximal ideals, quotient rings and the first ring isomorphism theorem.
• Classes of rings and fields, including matrix rings, Euclidean domains, rings of polynomials and fields of fractions of integral domains.
• Basic theory of field extensions, including degrees of extensions, finite extensions, simple extensions, algebraic elements, transcendental elements, algebraic extensions, adjoining subfields and splitting fields.

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