Galois Theory (MATH10080) Spring 2015
Update (November 2019): The links broke when my Edinburgh account closed. At the bottom of the page you may find the relevant materials in (merged) pdfs. However, the slides for my lectures are not provided.
Welcome to the course page for Galois Theory (Math 10080) Spring 2015; see the DRPS page for course objectives and timetable On this page, you will be able to find slides from the lectures as well as course notes; there is considerable overlap between the two. I am indebted to Tom Lenagan whose notes I used as the basis Notes 1-3 of mine. Here is the Galois Theory Syllabus.
Slides
- Motivation for Galois Theory
- Lecture 1 slides: Rings and Fields
- Lecture 2 slides: Ring Isomorphism Theorems
- Lecture 3 slides: Fields
- Lecture 4 slides: Irreducible polynomials
- Lecture 5 slides: Roots of polynomials
- Lecture 6 slides: Splitting fields
- Lecture 7 slides: Degree, dimension and field extensions
- Lecture 8 slides: Field extension isomorphisms
- Lecture 9 slides: Splitting field isomorphisms
- Lecture 10 slides: Cubic extensions
- Lecture 11 slides: Cubic extensions continued
- Lecture 12 slides: Solving the cubic
- Lecture 13 slides: Galois groups
- Lecture 14 slides: Galois groups and splitting fields
- Lecture 15 slides: Normality
- Lecture 16 slides: The Galois Correspondence
- Lecture 17 slides: The Galois Correspondence continued
- Lecture 18 slides: Solvability 1
- Lecture 19 slides: Solvability 2
- Lecture 20 slides: Insolvability of the quintic
- Lecture 21 slides: Return to constructibility 1
- Lecture 22 slides: Return to constructibility 2
Preparing for the exam
- Follow this link to find past papers at the Univsersity of Edinburgh from 2004/5 - 2013/4. Log-in to EASE and search for 'Jewels of Algebra' course. Our exam is modelled on these papers and the assessments (you will not be examined on Exercises 2,3,4 on Assessment 1).
- Notes from the revision on 23/4/15.
Notes
- Notes 1: Rings and Fields
- Notes 2: Fields and Irreducible polynomials
- Notes 3: Field extensions and polynomials
- Notes 4: Dimension, degree and field extensions
- Notes 5: Cubic extensions and equations
- Notes 6: Automorphisms and splitting fields with splitting implies normality 2.pdf (updated 3/4/2015) 7 pages
- Notes 7: The Galois correspondence (posted 13/3/15) 7 pages
- Notes 8: Solvability and the Quintic (posted 24/3/15, updated 7/4/15) 14 pages
- Notes 9: Constructibility (3/4/15) 6 pages
- Appendix: Transitive subgroups of S_n for n <= 5
Additionally, here is an appendix on the Primitive Element Theorem. You will not be examined on this. I merely provide it because the PET can be a useful tool in the theory and this may clarify our characterization of Galois extensions.
Tutorials
- Tutorial 1: Constructibility handout & Euclid the Game
- Tutorial 2: Quadratic extensions
- Tutorial 3: Biquadratic extensions
- Tutorial 4: Quartic extensions
- Tutorial 5: The Galois Correspondence
- Tutorial 6: Constructibility - you should visit the following construction of the regular pentagon by Jennifer Silverman in GeoGebra.
Assessments and Solutions
- Assessment 1 and Solutions 1 due Wednesday, 28 January 2015
- Assessment 2 and Solutions 2 due Wednesday, 11 February 2015
- Assessment 3 and Solutions 3 due Wednesday, 4 March 2015
- Assessment 4 and Solutions 4 due Wednesday, 1 April 2015
- Assessment 5 and Solutions 5 cancelled (due to industrial action)
Note that there is no assignment due 15 March 2015 and that now we only have 4 assessments due. Additionally, the homework grade is calculated best 3 marks out of 4.
Here is the link to Dr. Sue Sierra's Group Theory course from semester 1. Students enrolled in this class should be comfortable with the material in her notes.
GaloisTheorySyllabus.pdfKHughes - Galois Theory Notes - Spring 2015.pdfKHughes - Galois Theory Assessments - Spring 2015.pdfKHughes - Galois Theory Tutorials - Spring 2015.pdfKHughes - Galois Theory Exams and Revision - Spring 2015.pdf