• Offered for Year: 2025/26

• Module Leader(s): Dr Thorsten Heidersdorf

• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus

Semesters

Your programme is made up of credits, the total differs on programme to programme.

Aims

To introduce the viewpoint of modern algebra by extending the work of earlier modules and to provide a basis for further study of algebra.

Module summary

Modern algebra deals with sets equipped with operations resembling some or all of addition, subtraction, multiplication and division.

One example of such a system is a group which is equipped with only a single binary operation. Groups encapsulate a mathematical notion of symmetry. Groups will be introduced, and their basic properties will be studied via permutation groups, building on knowledge of mappings and permutations.

Many mathematical sets come with two operations which mimic addition and multiplication. Examples of this are fields and rings. We study examples of finite fields. We consider polynomials defined over a field and means of determining reducibility or irreducibility in some cases. We find that polynomials over a field form a ring and we consider the possibility of factorisation in a ring. In particular, we study rings consisting of complex numbers and see examples of such rings in which some numbers have more than one irreducible factorisation.

Outline Of Syllabus

Groups: definition, properties, examples. Symmetric, alternating and dihedral groups. Subgroups, conjugation, homomorphisms, isomorphisms, cosets, Lagrange's Theorem.

Multiple examples of rings and fields, including polynomial rings, Zn, and complex number rings. Applications to the study of polynomials, their roots and their factorisation. Factorisation in rings.

Teaching Methods

Teaching Activities

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Module drop-in sessions allow students to receive learning support in areas where they may need additional guidance.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams

Other Assessment

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

Reading Lists

• MAS2703's Reading List

Timetable

• Timetable Website: www.ncl.ac.uk/timetable/
• MAS2703's Timetable