	Department of Mathematics & Computer Science

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MA2161 Algebra II

MA2161(=MC255) Algebra II

Credits: 10

Convenor: Dr. A. Baranov

Semester: 2

Prerequisites:	essential: MA2102(=MC241), MA2111(=MC254)
Assessment:	Coursework: 20%	One and a half hour exam: 80%

Lectures:	18	Problem Classes:	5
Tutorials:	none	Private Study:	52
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	75

Explanation of Pre-requisites

Essential for this module is the theory of groups and rings developed in MA2111, including Euclidean domains and their ideal structure, quotient rings and abelian groups. The concepts of linear maps and matrices and change of basis from MA1152 and MA2102 are also needed.

Course Description

In this course, we continue the development of the theory of rings and modules which began in the module MA2111. It is possible to classify all finitely-generated abelian groups; each such group can be written as a direct product of copies of ${\bf Z}$ and cyclic groups. Secondly, it is possible, by means of a change of basis, to bring any matrix over the complex numbers into a very nice form, called the Jordan canonical form.

Both of these theorems are aspects of a single classification theorem which provides a description of all finitely-generated modules over a principal ideal domain (PID). The classification of finitely-generated abelian groups is obtained by specialisation to the case of finitely-generated modules over the PID ${\bf Z}$ (which is what finitely-generated abelian groups are). The Jordan canonical form for matrices is obtained by considering finitely-generated modules over the PID ${\bf C}[x]$ .

The main aim of the course is to prove this classification theorem and to consider these applications. In order to discuss the theorem, it is necessary to develop the theory of modules further. We need to consider the concepts of submodules and factor modules, and we need to generalise the idea of a basis of a vector space to the module case. We then consider the proof of this theorem for Euclidean domains (EDs), which involves proving that any matrix with entries in an ED can be reduced to a diagonal matrix with entries $d_1,d_2,\ldots ,d_n$ on the diagonal, where $d_1\vert d_2,\ d_2\vert d_3,\ d_3\vert d_4, \ldots ,d_{n-1}\vert d_n$ , using elementary row and column operations. Finally, we consider the application of the theorem to the classification of finitely-generated abelian groups (which is obtained by studying modules over the PID ${\bf Z}$ ) and the normal forms for matrices over a field (which are obtained by studying the PID ${\bf F}[x]$ , for ${\bf F}$ a field).

Further Study

Topics related to the course are covered in the material mentioned in the `Reading' section below. This module will provide students with a good understanding of abstract algebra, which is an important language of advanced mathematics, as well as the important classification of finitely-generated modules over a principal ideal domain. Further study of rings, and the relationship to field theory, is possible in the third level module MA3101, Abstract Algebra, and groups can be studied further in the third level module MA3131, Group Theory. Groups and rings also play a role in a number of level modules.

Aims

The aim of this course is to study the theorem classifying the finitely-generated modules over a principal ideal domain, to give a proof for Euclidean Domains, and to consider its applications to two important special cases: the classification of finitely-generated abelian groups, and normal forms for matrices over a field.

Objectives

To know the definitions of and understand the key concepts introduced in this mo dule.

To understand and be able to use the main results and proofs of this course.

To understand the definition and the basic properties of modules.

To understand that every module is the homomorphic image of a free module.

To understand the classification of finitely-generated modules over a principal ideal domain, and how this can be applied to the case of finitely-generated abelian groups.

To apply this classification to the case of a linear operator on a vector space, in order to determine its rational canonical form and (in the case where the field is the complex numbers) its Jordan canonical form.

Transferable Skills

The ability to understand abstract ideas and construct rigorous logical arguments.

The ability to determine wheter a proof is correct.

The ability to solve mathematical problems.

The ability to present written arguments and solutions in a coherent and logical form.

Syllabus

Group Concepts Abelian group, cyclic group, generators of a group, finitely-generated group.

Module Concepts Module, submodule, module homomorphism, kernel, image, factor module, cyclic module, generators of a module, finitely-generated module, free module, rank of a free module, direct sum of modules, elementary transformations of matrices, invariant factor matrix, rational canonical form, Jordan canonical form.

Theorems Every module is a homomorphic image of a free module, Classification of finitely generated modules over a PID, Application to finitely generated abelian groups, Application to the rational canonical form for matrices over a field, and the Jordan normal form for matrices over the complex numbers.

Reading list

Background:

I. Herstein, Topics in Algebra, Wiley.

Details of Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the Midsummer exam period. The 20% coursework contribution will be determined by students' solutions to four sets of work.

The exam paper will contain 4 questions. Any number of questions may be attempted, but only the best 3 answers will be taken into account. Full marks may be obtained for answers to 3 questions. All questions will carry equal weight.