Algebra and Analysis

• Credit value: 30 credits at Level 6
• Convenor: Professor Steven Noble
• Assessment: short written assignments (20%) and a two-hour examination (80%)

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

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.