Algebra and Analysis
Overview
- Credit value: 30 credits at Level 6
- Convenor: Professor Steven Noble
- Assessment: short written assignments (20%) and a two-hour examination (80%)
Module description
In this module you will learn key concepts from abstract algebra and analysis. The module builds on the Level 5 modules Algebra 2 and Calculus 2. One half covers material on algebraic structures such as groups, rings, fields and vector spaces, giving you a thorough grounding in these topics. The other half provides the theoretical background to calculus, formally defining the idea of a limit, continuity and differentiability of a function and the Riemann integral.
The learning materials for this module comprise a mixture of short instructional videos, online quizzes to test your understanding, face-to-face examples classes, full course notes, and exercises to try at home.
Indicative syllabus
Real variable
- Number systems: real numbers; supremum and infimum; epsilon notation
- Sequences: divergence and convergence; limits; subsequences
- Series: divergence and convergence; tests for divergence and convergence
- Functions: limits; continuity; tests for continuity and discontinuity
- Differentiation: Rolle’s Theorem; Mean Value Theorem; l’Hôpital’s rule
- Integration: Riemann integral; lower and upper Riemann sums
- Power series: Taylor polynomial; Taylor’s Theorem
Linear algebra
- Brief revision of linear algebra from Algebra 2
- Vector spaces over arbitrary fields
- Linear transformations and their properties
Groups
- Brief revision of group theory from Algebra 2
- Normal subgroups and quotients
Rings, fields and polynomials
- Definitions and examples of rings and fields including polynomial rings
- The division algorithm and the remainder theorem for polynomials
- Characteristic of a field and use of polynomials to construct finite fields
Learning objectives
By the end of this module, you will be able to:
- understand and use techniques from abstract algebra and analysis
- apply a range of results from abstract algebra and analysis
- construct mathematical arguments to establish a range of mathematical results
- understand the importance of assumptions and the possible consequences of their violation
- appreciate the power of generalisation and abstraction in the development of mathematical theories
- demonstrate a deeper knowledge of abstract algebra and analysis
- comprehend conceptual and abstract material
- demonstrate a logical and systematic approach to problem solving
- transfer knowledge and expertise from one context to another, by applying mathematical techniques in unfamiliar situations.