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MA2161 Algebra II

MA2161 Algebra II

Credits: 10 Convenor: Dr. A. Baranov Semester: 2 (weeks 15 to 26)
Prerequisites: essential: MA2102, MA2111
Assessment: Coursework: 20% Examination: 80%
Lectures: 18 Problem Classes: 5
Tutorials: none Private Study: 47
Labs: none Seminars: none
Project: none Other: none
Surgeries: 5 Total: 75

Subject Knowledge

Aims

The aim of this course is to prove the theorem classifying the finitely-generated modules over a principal ideal domain, and to consider its applications to the classification of finitely-generated abelian groups and canonical forms of matrices.

Learning Outcomes

Students should

$\bullet$ know the definitions of and understand the key concepts introduced in this course;

$\bullet$ understand and be able to use the main results and proofs of this course;

$\bullet$ understand the definition and the basic properties of modules;

$\bullet$ understand that every module is the homomorphic image of a free module;

$\bullet$ 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;

$\bullet$ 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.

Methods

Lectures and problem classes, handouts.

Assessment

Marked problem sheets, written examination.

Subject Skills

Aims

To develop problem solving skills, written communication skills.

Learning Outcomes

Students will be able to use the techniques taught within the module to solve problems and be able to present written arguments in a coherent and logical form.

Methods

Class sessions, coursework, exam.

Assessment

Marked problem sheets, exam.

Explanation of Pre-requisites

Essential for this course 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 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).

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

Recommended:

M. Artin, Algebra, Prentice Hall.

P. Cohn, Algebra, Wiley.

B. Hartley and T. Hawkes, Rings, Modules and Linear Algebra, Chapman and Hall.

Background:

I. Herstein, Topics in Algebra, Wiley.

Resources

Problem sheets, handouts, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA2201 Introductory Statistics Up: ModuleGuide03-04 Previous: MA2151 Abstract Analysis

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Last updated: 2004-02-21
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