MT3505 Algebra: Rings and Fields

Academic year

2024 to 2025 Semester 2

Further information on which modules are specific to your programme.

Key module information

SCOTCAT credits

15

The Scottish Credit Accumulation and Transfer (SCOTCAT) system allows credits gained in Scotland to be transferred between institutions. The number of credits associated with a module gives an indication of the amount of learning effort required by the learner. European Credit Transfer System (ECTS) credits are half the value of SCOTCAT credits.

SCQF level

SCQF level 9

The Scottish Credit and Qualifications Framework (SCQF) provides an indication of the complexity of award qualifications and associated learning and operates on an ascending numeric scale from Levels 1-12 with SCQF Level 10 equating to a Scottish undergraduate Honours degree.

Planned timetable

11.00 am Mon (odd weeks), Wed & Fri

This information is given as indicative. Timetable may change at short notice depending on room availability.

Module coordinator

Dr T D H Coleman

Dr T D H Coleman
This information is given as indicative. Staff involved in a module may change at short notice depending on availability and circumstances.

Module description

This module builds on the study of Abstract Algebra which was begun in MT2505. It shows how various familiar mathematical structures such as the integers, the real numbers, sets of matrices and sets of polynomials can all be viewed as examples of a structure called a 'ring' which captures the property of having two operations (addition and multiplication). We develop theory to understand rings in general, which then enables us to better understand various properties of these examples - some familiar and some new to us. We also take useful or interesting properties of the motivating examples (eg number theory as performed in the integers) and see how this can be meaningfully extended to other rings. In the process, we also briefly investigate a few other topics (for example the order theory of partially ordered sets) and see how this can be used to prove ring theory results.

Relationship to other modules

Pre-requisites

BEFORE TAKING THIS MODULE YOU MUST PASS MT2505

Assessment pattern

2-hour Written Examination = 90%, Coursework = 10%

Re-assessment

Oral examination = 100%

Learning and teaching methods and delivery

Weekly contact

2.5 hours of lectures and 1 tutorial.

Scheduled learning hours

35

The number of compulsory student:staff contact hours over the period of the module.

Guided independent study hours

115

The number of hours that students are expected to invest in independent study over the period of the module.

Intended learning outcomes

  • Appreciate how the definition of ring/field captures key properties of many familiar mathematical examples (e.g. number systems, polynomials, matrices); know and be able to work with the Ring Axioms
  • Be familiar with special types of ring (e.g. integral domain, division ring, field), special types of substructure (e.g. subring, ideal, principal/prime/maximal ideal) and quotient rings; be able to prove results about these
  • Be familiar with Zorn's Lemma and chain conditions for ideals (Noetherian and Artinian) and be able to prove relevant results
  • Understand concepts such as divisibility, associates, primes and irreducibles and hence the notion of unique factorization, and be able to work with these
  • Know the definition of norm and be able to perform norm calculations in appropriate rings
  • Be able to work with polynomial rings and fields of fractions; understand the importance of quotients of polynomial rings in constructing finite fields and know the Fundamental Theorem of Finite Fields