Galois Theory, 2015-2016


N. Dogra. Room: HG03.721, email: netandogra SYMBOL gmail SYMBOL com. He also has a website for this course.

Time and place

The lectures are on Wednesdays from 15:45 until 17:30 at HG00.308. Exercise classes are on Fridays from 10:45 until 12:30 at HG03.054. The lectures will be given in English, the exercise classes will be in Dutch.


This course will follow the Dutch book Galoistheorie by Frans Keune. This book can be ordered here. The material for the first weeks will be covered in chapters 1,3 and 4 of this book, which are available here.


Depending on the number of students, there will either be a written exam, or a larger homework assignment followed by an oral exam. To be eligible for participating in the exam, you need to have a sufficient mark (6 or higher) on at least 10 of the homework sets (see below).

The final mark will be rounded to a half-integer not equal to 5.5. The final mark may be rounded up by at most one point depending on the average mark for the homework assignments.


There will be a set of homework problems every week, which will be announced on this website after the lecture. The homework problems are to be handed in in the mailbox of Milan Lopuhaä the next Tuesday at 17:00. Exceptions to this schedule will be announced on the website. The exercises may be handed in in English or Dutch.

The marks are available on Blackboard.


Date Material Keune Homework
September 2 Recap of field theory: field extensions, algebraic and transcendental elements, homomorphisms of field extensions The non-asterisked parts of these exercises
September 9 Adjoining roots of polynomials, splitting fields of polynomials, properties of Hom-sets of such extensions end of chapter 1, first half of chapter 4 These exercises
September 16 normal extensions, Galois extensions second half of chapter 4, most of chapter 3 These exercises
September 23 normal extensions, separable extensions, Galois extensions and groups These exercises
September 30 separable extensions, primitive element theorem These exercises
October 7 properties of separability, fundamental theorem of Galois theory, cyclotomic extensions The non-asterisked parts of these exercises
October 14Galois group of cyclotomic fields The non-asterisked parts of these exercises
October 21 proof and historical motivation of the fundamental theorem These exercises
November 11 No lecture!No Exercise class!
November 18 Symmetric polynomials and their use in constructing Galois extensions The non-asterisked parts of these exercises
November 25 Radical extensions and solvable groups These exercises
December 2 non-solvability of the general quintic equation, Galois groups of extensions of Q, reductions mod p These exercises
December 9 small degree extensions of Q, Kummer theory, normal basis theorem The non-asterisked parts of these exercises
December 16 Last part of exam material, first part of infinite Galois theory The non-asterisked parts of these exercises
January 6
January 13

Glossary of Galois theory

English Dutch
algebraic closure algebraïsche afsluiting
compass and ruler passer en liniaal
constructible construeerbaar
cubic derdegraads
cyclotomic polynomial cyclotomisch polynoom, cirkeldelingsveelterm
extension uitbreiding
field lichaam (Nederland), veld (Vlaanderen)
polynomial polynoom, veelterm
quadratic tweedegraads
quartic vierdegraads
root nulpunt; wortel
solvable oplosbaar