| Credits: 10 | Convenor: Dr. R. J. Marsh | Semester: 1 (weeks 7 to 12) |
| Prerequisites: | essential: MC144 | |
| Assessment: | Coursework/laboratory project: 20% | One and a half hour exam: 80% |
| Lectures: | 18 | Classes: | none |
| Tutorials: | 5 | Private Study: | 46 |
| Labs: | 6 | Seminars: | none |
| Project: | none | Other: | none |
| Total: | 75 |
), 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 x2 = 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 x2 + 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.
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.
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.
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.
Congruence as an equivalence relation, residue, Zn, 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 Zp.
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.
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.