Monash University Handbook 2011 Undergraduate - Unit

Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.

Synopsis

Rings, fields, ideals, algebraic extension fields. Coding theory and cryptographic applications of finite fields. Gaussian integers, Hamilton's quaternions, Chinese Remainder Theorem. Euclidean Algorithm in further fields.

Objectives

At the completion of this unit, students will be able to demonstrate understanding of advanced concepts, algorithms and results in number theory; the use of Gaussian integers to find the primes expressible as a sum of squares, Diophantine equations, primitive roots; the quaternions, the best known skew field; many of the links between algebra and number theory; the most commonly occurring rings and fields: integers, integers modulo n, rationals, reals and complex numbers, more general structures such as algebraic number fields, algebraic integers and finite fields; and will have developed skills in the use of the Chinese Remainder Theorem to represent integers by their remainders; performing calculations in the algebra of polynomials; the use of the Euclidean algorithm in structures other than integers; constructing larger fields from smaller fields (field extensions); applying field theory to coding and cryptography.

Assessment

Examination (3 hours): 70%
Assignments and tests: 30%

Chief examiner(s)

Dr Ian Wanless

Contact hours

Three 1-hour lectures and an average of one 1-hour support class per week

Prerequisites

MTH2122, MTH2121, MTH3121 or MTH3122

Co-requisites

MTH2122, MTH2121, MTH3121 or MTH3122