Mathematics

ECM3738 - Galois Theory (2015)

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MODULE TITLEGalois Theory CREDIT VALUE15
MODULE CODEECM3738 MODULE CONVENERProf Nigel Byott (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
Number of Students Taking Module (anticipated) 10
DESCRIPTION - summary of the module content

Drawing on key ideas in the theory of groups and fields, you will learn core elements of the theory of field extensions.  You are already familiar with the idea that the real numbers can be extended to the complex numbers by introducing a new number as the square root of -1; Galois theory formalises such constructions and explores the intriguing relationship between groups and field extensions.

 

As an important application of Galois theory, you will understand why there can be no algebraic solution to the general quintic polynomial with rational coefficients.

 

Prerequisite module: ECM2711 and ECM2712 or equivalent

AIMS - intentions of the module

The aim of this module is to motivate and develop Galois theory both as an abstract theory and through the study of important applications.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:
 

Module Specific Skills and Knowledge:
1 state and apply key definitions in Galois theory;
2 state, prove and apply core theorems in Galois theory.
Discipline Specific Skills and Knowledge:
3 perform computations accurately;
4 use abstract reasoning to solve a range of problems.
Personal and Key Transferable / Employment Skills and Knowledge:
5 communicate your findings effectively in writing;
6 work independently and manage your time and resources effectively.

 

SYLLABUS PLAN - summary of the structure and academic content of the module

- review of the field axioms, the characteristic of a field, examples. Field extensions, degree, finite and algebraic extensions, extensions obtained by adjoining a root of an irreducible polynomial, degree in a tower of extensions;
 

- splitting fields and algebraic closure. Separable and inseparable extensions. Cyclotomic polynomials and extensions. Automorphisms of a field. The group of automorphisms, the fixed field of a subgroup of automorphisms, the Galois correspondence. The fundamental theorem of Galois theory. Finite fields. Finite extensions of finite fields. The Galois theory of finite fields. Composite extensions and simple extensions. The primitive element theorem;
 

- cyclotomic extensions and abelian extensions over Q. Abelian groups as Galois groups over Q. Cyclic extensions and Kummer theory. Galois groups of polynomials. Solvable and radical extensions: solution of cubic and quartic equations by radicals, insolvability of the quintic. Computation of Galois groups over Q. Hilbert's irreducibility theorem. Polynomials with Galois groups Sn and An.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33.00 Guided Independent Study 117.00 Placement / Study Abroad 0.00
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 33 Lectures including example classes
Guided independent study 117 Lecture and assessment preparation; wider reading
     

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Exercises One sheet fortnightly (or equivalent) All Verbal and generic feedback in example classes
       
       
       
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Exam mark – verbal feedback on request
Coursework – based on questions submitted for formative assessment 20 30 hours in total All Annotated script and written feedback
         
         
         

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All above Written exam (100%) All August Ref/Def period
       
       

 

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: College to provide hyperlink to appropriate pages

 

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Stewart, I. Galois Theory Chapman and Hall 2004 [Library]
Set Rotman, J. Galois Theory Springer 1998 [Library]
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES ECM2711, ECM2712
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 6 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Friday 09 January 2015 LAST REVISION DATE Friday 09 January 2015
KEY WORDS SEARCH Galois; field; extension; group; polynomial.