# GALOIS THEORY - 2018/9

Module code: MAT3011

Module Overview

Galois Theory applies the principles of algebraic structure to questions about the solvability of polynomial equations. The feasibility of certain geometrical constructions is also considered.

Module provider

Mathematics

FISHER D Dr (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 6

JACs code: G100

Module cap (Maximum number of students): N/A

Module Availability

Semester 2

Prerequisites / Co-requisites

MAT2048 Groups and Rings

Module content

Indicative content includes:

Theory of polynomials.  Criteria for irreducibility.
Solution of cubic and quartic equations.
Field extensions. The degree of an extension.  The Tower law.
Geometric constructions.
Field automorphisms and their properties.
The Galois correspondence and the fundamental theorem.
Solvable groups.  Conditions for solvability of polynomials by radicals.

Assessment pattern

Assessment type Unit of assessment Weighting
Examination EXAMINATION (2 HOURS) 80
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 20

Alternative Assessment

N/A

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate their ability to

·         construct and interpret mathematical arguments in the context of this module;

·         display subject knowledge by recalling key definitions and results;

·         apply the techniques learnt to both routine and unfamiliar problems.

Thus, the summative assessment for this module consists of:

·         One two-hour examination at the end of Semester 1, worth 80% of the module mark.

·         In-semester test, worth 20% of the module mark.

Formative assessment and feedback

Students receive written comments on their marked coursework assignments.  Maple TA may be used in assignments to provide instant grading and immediate feedback.  Verbal feedback is provided in lectures and office hours.

Module aims

• review the theory of groups, rings, fields and polynomials.
• develop and apply the theory of field extensions and Galois groups.
• show the power of abstract algebra to produce practical and applicable results.

Learning outcomes

Attributes Developed
1 Demonstrate a deeper appreciation of algebraic structures and of the power of linking different structures   KCT
2 Solve cubic and quartic equations, understanding the limitations of the methods KC
3 Evaluate the degree of finite field extensions and apply this to algebraic and geometric examples KC
4 Evaluate specific Galois groups and relate their structure to that of field extensions and to the solvability of polynomial equations . KC
5 Construct simple proofs similar to those encountered in the module. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Independent Study Hours: 117

Lecture Hours: 33

Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

An awareness of the applicability of abstract algebra to classical problems.
Knowledge of the historical context in which the subject material was developed.
Experience of the methods used to interpret, understand and solve problems in Galois theory.

The learning and teaching methods include:

Three 50-minute lectures per week for eleven weeks, some being used as tutorials, problem classes and in-semester tests.
Online notes supplemented by additional examples in lectures.
Two unassessed coursework assignments, marked and returned.
Personal assistance given to individuals and small groups in office hours.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list for GALOIS THEORY : http://aspire.surrey.ac.uk/modules/mat3011

Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2018/9 academic year.