# MTH3150 - Algebra and number theory II

## 6 points, SCA Band 0 (NATIONAL PRIORITY), 0.125 EFTSL

### Undergraduate Faculty of Science

#### Leader(s): Dr Ian Wanless

#### Offered

Clayton Second semester 2009 (Day)

#### 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%

#### Contact hours

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

#### Prerequisites

MTH2122, MTH2121, MTH3121 or MTH3122