# CANCELLED - Euclidean Rings

### Course content

The customary definition of a Euclidean ring is that it is an integral domain that has a Euclidean algorithm taking values in the set of natural numbers N. In this course we will define a Euclidean ring more generally as a commutative ring for which there is a Euclidean algorithm taking values not necessarily in N but in any well-ordered set. In the first part of the course we shall discuss general properties of Euclidean rings. Among others we will discuss the notion of the smallest algorithm and its transfinite construction via Motzkin's sets, which will lead to a criterion for a ring to be Euclidean. The second part of the course will be devoted to euclidianity of rings of algebraic integers of number fields with the focus on the quadratic case. In particular, we will see examples of rings which are Euclidean but not norm-Euclidean and examples of principal ideal domains which are not Euclidean for any algorithm. We shall also discuss Euclidean minima of number fields and various results concerning them.

Education

MSc programme in Mathematics

Learning outcome

Knowledge: After completing the course the student will know the subjects mentioned in the description of the content.

Skills: At the end of the course the student is expected to be able to follow and reproduce arguments at a high, abstract level corresponding to the contents of the course.

Competencies: At the end of the course the student is expected to be able to apply abstract results from the curriculum to the solution of concrete problems.

Weekly: 4 hours of lectures and 3 hours of exercises for 8 weeks.

Algebra 2 (Alg2) or similar.

ECTS
7,5 ECTS
Type of assessment
Continuous assessment
Three mandatory homework assignments. Each assignment can be submitted only once. each count one third of the grade
Aid
All aids allowed
Marking scale
Censorship form
No external censorship
One internal examiner
##### Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 32
• Exercises
• 24
• Exam Preparation
• 40
• Preparation
• 110
• English
• 206