Galois Theory

Galoistheorie

Course manual 2023/2024

Course content

Galois theory is one of the central and classical aspects of mathematics, focusing on the symmetries of fields and the roots of algebraic equations. Through Galois theory, several classical problems in mathematics can be solved. We cover elementary Galois theory and demonstrate how these aforementioned classical problems are resolved.

Study materials

Syllabus

•  Stevenhagen - Lecture notes Algebra 3. Available on canvas

Other

• These are published in the lecture notes, and on canvas

Objectives

• The student has elementary knowledge of Galois theory.
• The student is able to explain why there does not exist a general formula to solve polynomials of degree 5 and up
• The student can explain why it is possible to construct the regular 17-gon with ruler and compass, and why you cannot construct the regular 19-gon.
• The student can recognise and prove that the Galois group of the polynomial f=X^4−X^2−2∈Q[X] is isomorphic with Klein's four group.
• The student is able to determine all subfields of Q(ζ_15) where ζ_15 is a primitive r15-th root of unity, by giving for each such subfield a primitive element.
• The student can recognise and prove that the roots of the polynomial f=X^4−X^3+X^2−X+1∈Q[X] in C can be constructed with ruler and compass.

Teaching methods

• Lecture
• Self-study
• exercise session

Learning activities

Attendance

Programme's requirements concerning attendance (OER-B):

• Each student is expected to actively participate in the course for which he/she is registered.
• If a student cannot be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
• It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
• In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
• In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, it can be decided to deny the positive effects of homework to be counted towards the final grade of the course. In case of personal circumstances, as described in OER-A Article A-6.4, a different arrangement will be proposed in consultation with the study advisor.

Additional requirements for this course:

Attendance at the tutorials is mandatory. If you have not attended at least 80% of the tutorials, you forfeit your right to the resit exam, as stated in OER-B Article 4.9, Section 2.

Assessment

If you do not take a resit, then the homework counts for 20%, the midterm exam for 30%, and the final exam for 50% in the final grade.

If you take a resit, both the grade for the final exam and the grade for the midterm exam are discarded and replaced by the resit grade (even if the resit grade is lower!). The homework grade, however, still counts. So, in this case, the distribution is 20% homework and 80% resit grade.

Assignments

There will be weekly homework assignments. These are published on canvas.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Honours information

For students who score at least an 8 on the midterm exam, it is possible to participate in the honors program. This honors program focuses on infinite Galois theory.

Contact information

Coordinator

• Arno Kret