Undergraduate Course: Group and Galois Theory (MATH10078)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 10 (Year 4 Undergraduate) 
Availability  Available to all students 
SCQF Credits  20 
ECTS Credits  10 
Summary  This is a course in abstract algebra, although connections with other fields will be stressed as often as possible. It begins with a systematic study of the basic structure of groups, finite and infinite. There will also be some ring theory. Then it will cover some of the jewels in the crown of undergraduate mathematics, drawing together groups, rings and fields to solve problems that resisted the
efforts of mathematicians for many centuries. The powerful central ideas of this course are now crucial to many modern problems in algebra, differential equations, geometry, number theory and topology. 
Course description 
 Group actions & Sylow Theorems
 Homomorphisms, isomorphisms & factor groups
 Group Presentations
 Simple groups & Composition Series
 Polynomial rings & finite fields
 Classification of finite abelian groups
 PIDs & their modules
 Fields: examples, constructions and extensions
 Separability, normality & splitting fields
 Field automorphisms & Galois groups
 The fundamental theorem of Galois Theory
 Solvable groups and the insolubility of the general quintic
 Ruler and Compass constructions
 Calculation of Galois groups
 Transcendence

Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information
Not being delivered 
Learning Outcomes
1. Facility with the Sylow theorems, group homomorphisms and presentations, and the application of these in order to describe aspects of the intrinsic structure of groups, both abstractly and in specific examples.
2. Ability to manipulate composition series, through both the proof of abstract structural properties and the calculation of explicit examples.
3. Capacity to work with the classes of rings and fields appearing in the course, particularly specific calculations around finite fields and polynomials.
4. Facility with fields and their extensions, including expertise in explicit calculations with and constructions of examples with various relevant desired properties.
5. Ability to handle Galois groups, abstractly and in explicit examples, by using a variety of techniques including the Fundamental Theorem of Galois Theory and presentations of fields.
6. Capacity to explain and work with the consequences of Galois Theory in general questions of mathematics addressed in the course, such as insolubility of certain classes of equations or impossibility of certain geometric constructions.
7. Be able to produce examples and counterexamples illustrating the
mathematical concepts presented in the course.
8. Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.

Reading List
Recommended :
 T S Blyth and E S Robertson, Groups (QA171.Bly )
 J F Humphreys, A Course in Group Theory (QA177 Hum)
 M A Armstrong, Groups and Symmetry (QA171 Arm )
 J J Rotman, The theory of groups: An introduction (QA171 Rot )
 J J Rotman, An introduction to the Theory of Groups (QA174.2 Rot )
Students are expected to have a personal copy of 'Galois Theory', Fourth Edition (Chapman and Hall / CRC) by Ian Nicholas Stewart. ISBN13: 9781482245820. 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  GGTh 
Contacts
Course organiser  Dr Susan Sierra
Tel: (0131 6)50 5070
Email: S.Sierra@ed.ac.uk 
Course secretary  Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk 

