6 EC
Semester 2, period 4, 5
5334TINT6Y
| Owner | Master Mathematics |
| Coordinator | Chris Lazda |
| Part of | Master Mathematics, year 1 |
This course will be an introduction to global class field theory, which is the study of abelian extensions of number fields. (An abelian extension is a Galois extension with abelian Galois group.) The main theorems of class field theory classify all abelian extensions of a given number field K, calculate their Galois groups, and describe the splitting behaviour of prime ideals in such extensions.
Over the rational numbers Q, class field theory is a consequence of the Kronecker-Weber theorem, and we will start the course by exploring various aspects of this result, including connections with classical results in number theory such as quadratic reciprocity, and the representability of primes by quadratic forms. We will then move on to general number fields, using techniques from both analytic number theory and abstract algebra to prove Artin reciprocity, which describes the Galois group of a given abelian extension of number fields in terms of the Artin symbol. Finally, we will discuss the existence of so-called ray class fields, which are generalisations of the cyclotomic extensions of Q and provide a 'sufficient' supply of abelian extensions of a given number field K.
There will also be computer lab sessions involving explicit and computational aspects of algebraic number theory and class field theory.
Students wishing to attend this course will need to have attended the Algebraic Number Theory Mastermath course in the first semester, or to have a good understanding of the contents of this course.
Students will need their own laptops to participate in the lab sessions.
By the end of the course, the students will understand the statements and proofs of the main theorems of global class field theory. They will be able to compute simple examples, and use the general theory they have learned to solve particular Diophantine problems. They will also be able to use computer algebra packages to make explicit calculations in algebraic number theory.
Classes will involve lectures covering the basic theory, including examples and applications. Students will spend self-study time on problem sheets designed to test and enhance their understanding of the course material. These will then be discussed during class. Familiarity with computer algebra packages will be developed during two computer lab sessions, led by Prof. Dokchitser.
|
Activity |
Hours |
|
|
Hoorcollege |
22 |
|
|
Computational aspects sessions |
8 |
|
|
Tentamen |
3 |
|
|
Self study |
135 |
|
|
Total |
168 |
(6 EC x 28 uur) |
The programme does not have requirements concerning attendance (OER-B).
Additional requirements for this course:
Students should inform the course coordinator of any absence.
| Item and weight | Details |
|
Final grade | |
|
60% Tentamen | Must be ≥ 5 |
|
20% Computational Assignment | |
|
20% Homework |
The exam will be a written exam with no calculator or notes allowed. Students are required to obtain a minimum of 5 on the final exam involve pass the course. Any homework assignment handed in more than a week late will receive no marks. The computational assignment will need to be handed in at the final class of the semester (13/5).
Weekly homework problems will be returned to students and can be discussed in class, after they have been marked. Grades for the computational assignment can be discussed during office hours that will be communicated to the students in class.
There will be weekly problem sheets, which the students are expected to complete in their own time, either singly or in groups. These will be discussed at the start of the next class, and one (specified) problem each week will need to be handed in as marked homework, which will count towards the final grade.
After the two computer lab sessions the students will be given an assignment that they are expected to complete by the end of the course, to be handed in at the final class.
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
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The schedule for this course is published on DataNose.