Unit information: Algebra 2 in 2013/14

Unit name	Algebra 2
Unit code	MATH21800
Credit points	20
Level of study	I/5
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Walling
Open unit status	Not open
Pre-requisites	MATH11511 Number Theory & Group Theory and MATH 11005 Linear Algebra & Geometry
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

The overall aim of the unit is to develop the theory of commutative rings, and to apply it to solving problems concerning the factorisation of polynomials, the two squares theorem, algebraic numbers, ruler-and-compass constructions, and the construction of finite fields.

Many important objects in mathematics can be thought of as number-systems in the sense that they have both addition and multiplication. Examples include the integers, the rational numbers, the complex numbers, integers mod(n), polynomials, etc. We shall study some of these systems with the two aims of proving general results and immediately applying the theory to solve problems. For instance we shall consider unique factorisation in such systems, and use the uniqueness of factorisation of Gaussian integers to prove Fermat's assertion that every prime number of the form 4n + 1 is the sum of two squares.

Also, we shall develop the sort of mathematics which is used to show that certain geometric constructions are not possible, and we shall use it to prove the impossibility of dividing an arbitrary angle into three equal parts and, apart from one fact which we shall just have to quote, the impossibility of "squaring the circle". We shall also construct some examples of finite fields; this topic has now become very important because of its use in coding theory.

Intended Learning Outcomes

After taking this unit, students should be able to:

• state the basic definitions and results in the subject,
• utilise the fundamental proof techniques,
• solve problems similar to those worked in the lectures and set as homework.

Teaching Information

The course will be based on lectures and posted notes, with exercise and solution sheets. There will also be a problems class each week.

Assessment Information

The final assessment mark for the unit will be calculated as follows:

• 85% from a standard 2½ hour written examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted.
• 10% from five sets of homework questions;
• 5% from five in-class open-book 10 minute quizzes.

Reading and References

No particular text will be used, but advice will be given on further reading.