MA4FUA: Further Topics in Algebra

Module code: MA4FUA

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: 7

When you’ll be taught: Semester 2

Module convenor: Dr Basil Corbas , email: b.corbas@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST TAKE MA2ALA (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded: IN TAKING THIS MODULE YOU CANNOT TAKE MA3FUA OR TAKE MA3FTA OR TAKE MA3GAL (Compulsory)

Placement information: NA

Academic year: 2025/6

Available to visiting students: Yes

Talis reading list: No

Last updated: 3 April 2025

Overview

Module aims and purpose

This module is a sequel to the second year Algebra module, in which students were introduced to the concepts of groups and fields. 

This module 

• continues the study of finite groups: a number of interrelated concepts from this immense subject have been selected and put together in an elementary and coherent way;  
• applies prior knowledge of groups and fields to the beautiful subject of Galois Theory. 

Galois Theory examines the conditions under which polynomial equations can be solved using elementary algebraic operations. To a polynomial, whose roots we wish to determine, we associate a field and thereby a group. By studying the properties of this group, we determine the feasibility of finding the roots of the polynomial by elementary algebraic means. 

Students will see how these same methods can be employed to decide whether certain geometric constructions (like the squaring of a circle, or the duplication of a cube) can be carried out using only a straight edge and a compass. 

Module learning outcomes

By the end of the module students are expected to be able to: 

1. explain the theory of Sylow subgroups and apply it in various settings;  
2. explain and carry out the techniques used to classify groups of certain orders; 
3. explain theoretical aspects of Galois theory and apply it in explicit settings; 
4. explain the relationship between Galois theory and the problem of whether certain geometric constructions are possible by straight edge and compass 

Module content

The first half of the module studies in detail the theory of finite groups. In particular, the following topics will be discussed: 

• Introduction to the theory of G-sets: orbits, stabilizers, the class formula; 
• Applications to finite groups: normalizers, centralizers, p-groups and the Sylow theorems; 
• The group of automorphisms and of inner automorphisms of a group; 
• Semi-direct products and the holomorph of a group; 
• Finite Abelian groups. 

The second part of the module focuses on Galois Theory. In particular, the following topics will be discussed: 

• Finite extensions and algebraic extensions of fields; 
• The splitting field of a polynomial and its Galois Group over a given field; 
• The Galois correspondence and the fundamental theorem of Galois Theory;  
• Solvable groups and the solvability by radicals of a polynomial; 
• Geometric constructions by straight edge and compass. 

Structure

Teaching and learning methods

Lectures supported by tutorials. Learning materials (lecture notes/reading lists, tutorial problem sheets, assessments) made available via Blackboard. 

Study hours

At least 50 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

A mark of 50% overall 

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Reassessment

Additional costs