NMAK15012U Euclidean Rings
MSc programme in Mathematics
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.
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.
- Category
- Hours
- Exam Preparation
- 40
- Exercises
- 24
- Lectures
- 32
- Preparation
- 110
- Total
- 206
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- Credit
- 7,5 ECTS
- Type of assessment
- Continuous assessmentThree mandatory homework assignments. Each assignment can be submitted only once. each count one third of the grade
- Aid
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
One internal examiner
- Re-exam
30 minutes oral examination with 30 minutes time for preparation. All aids allowed during the preparation time, no aids allowed during the examination
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.
Course information
- Language
- English
- Course code
- NMAK15012U
- Credit
- 7,5 ECTS
- Level
- Full Degree Master
- Duration
- 1 block
- Placement
- Block 3
- Schedule
- A
- Course capacity
- No limit
- Continuing and further education
- Study board
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Course responsibles
- Piotr Aleksander Maciak (6-5064666c646e437064776b316e7831676e)