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Next: MC146 Introductory Real Analysis Up: Year 1 Previous: MC144 Proof and Logical Structures

MC145 Algebraic Structures and Number Systems

MC145 Algebraic Structures and Number Systems

Credits: 10 Convenor: Dr Dietrich Notbohm Semester: 1 (weeks 7 to 12)
Prerequisites: essential: MC144
Assessment: Coursework/laboratory project: 20% One and a half hour exam: 80%
Lectures: 18 Problem Classes: none
Tutorials: 5 Private Study: 46
Labs: 6 Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Explanation of Pre-requisites

This course uses the concept of proof and the proof strategies introduced in MC144 to investigate properties of number systems. Basic set theory, the properties of the natural numbers and the notion of an equivalence relation are also required from MC144.

Course Description

This course studies the properties of familiar number systems by investigating the extent to which we are able to solve polynomial equations. Starting with the integers, it is clear that the equation $2x=3$ has no integer solution, but this leads to the concept of divisibility with which to investigate integer equations of the form $ax = b$ for which there is an integer solution. However, in order to solve all equations of the form $ax = b$ where $a$ and $b$ are integers ($a \neq 0$), we need `more numbers' than just the integers, so we formally construct the rational numbers to enable us to solve these equations. Moving on, what about other equations like $x^2 = 2$? How do we solve this using only the rationals? The answer is that there is no solution which is a rational number, so again we need to construct some `more numbers'. We discuss the real numbers here but leave the precise definition to the module MC146. But even the real numbers are not enough, as the equation $x^2 + 1 = 0$ still poses a problem. So, finally, we construct the complex numbers and discuss the fundamental theorem of algebra which tells us that we now have `enough numbers' with which to solve all polynomial equations.

Many of these number systems share the same properties. For example, they all involve the familiar concepts of addition and multiplication. We end the course by setting, and hence unifying, these number systems in the formal framework of the algebraic structures of the field and ring.

Aims

This course studies the properties of familiar number systems and polynomials through the solution of equations and relates these number systems to the algebraic structures of the group, ring and field. The course aims to show that apparently different structures may share the same properties and thus motivate understanding of abstract axiomatic mathematics. We continue the emphasis on writing simple proofs and the proper use of mathematical notation from MC144. Maple is taught alongside the lecture course as a general tool for mathematics and this enables students to investigate more complex problems related to the lectured material than could otherwise be considered.

The course also aims to understand the interrelationships between the natural numbers, integers, rationals, real numbers and complex numbers, and the extent to which polynomial equations can be solved within each of these number systems.

Objectives

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

To understand, reconstruct and apply the main results and proofs covered in this course.

To calculate the greatest common divisor of two integers and of two polynomials over a field.

To understand and be able to use congruence arithmetic to solve a variety of problems.

To know the concepts of a field, ring and group and their relationships with the specific number systems and polynomial rings of this module.

To use Maple interactively and to solve problems involving curve sketching and solutions of equations.

Transferable Skills

The development of abstract mathematics and the axiomatic method.

The ability to apply taught principles and concepts to new situations.

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

The ability to use the techniques taught within the course to solve problems.

Knowledge of Maple as a tool for doing mathematics and specifically for curve sketching and solving equations.

Syllabus

The integers, division, greatest common divisor, division algorithm, Euclidean algorithm, coprime and prime integers and their properties.

Congruence as an equivalence relation, residue, Z$_n$, congruence arithmetic, cancellation, solving equations using congruence arithmetic including use of Euclidean algorithm.

The rational numbers constructed by an equivalence relation, field, ordered field, discussion of the real numbers and the least upper bound axiom, the finite fields Z$_p$.

Definition of a polynomial over a field, degree, leading coefficient, monic polynomial, greatest common divisor of two polynomials, division algorithm, Euclidean algorithm.

Algebraic form of a complex number, real and imaginary parts, modulus, conjugate, basic properties, Argand diagram, triangle inequality, solving polynomials and the statement and applications of the fundamental theorem of algebra, polar form of a complex number, argument, properties involving multiplication and division of complex numbers in polar form, De Moivre's Theorem, roots of unity.

Definition of a ring, commutative ring, unit, relationship between ring and field, examples, properties of the set of units of a ring, group, abelian group, relation to the additive structure of a ring, examples, Fermat's Little Theorem, example of a non-abelian group (using permutations).

The course also teaches the interactive use of the software package Maple as a tool in mathematics, and its use in investigating properties of the algebraic systems covered in the lectures. Topics covered with Maple include sketching the curve of a rational function, finding the roots of a polynomial, congruences and complex numbers.

Reading list

Background:

R. B. J. T. Allenby, Rings, Fields and Groups, 2nd. Ed., Arnold.

J. R. Durbin, Modern Algebra: an Introduction, 3rd Ed., Wiley.

A. P. Hillman & G. L. Alexanderson, Abstract Algebra: a First Undergraduate Course, 5th Ed., PWS-Kent.

K. E. Hirst, Numbers, Sequences and Series, Arnold.

W. K. Nicholson, Introduction to Abstract Algebra, PWS-Kent.

I. Stewart & D. Tall, The Foundations of Mathematics, Oxford University Press.

Details of Assessment

The final assessment of this module will consist of 10% coursework, 10% for a MAPLE project and 80% from a one and a half hour examination during the January exam period. The 10% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.

Next: MC146 Introductory Real Analysis Up: Year 1 Previous: MC144 Proof and Logical Structures

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Last updated: 2001-09-20
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