Algebraic Number Theory - M1 - 8EC

Aim of the course
The course provides a thorough introduction to algebraic number theory: introduction to algebraic numbers and number rings, ideal factorization, finiteness results on class groups and unit groups, explicit computation of these objects. We will also discuss algorithmic aspects using the system Sage (http://sagemath.org, http://cocalc.com).

 

Prerequisites
Undergraduate algebra, i.e., the basic properties of groups, rings, and fields, including Galois theory. This material is covered in first and second year algebra courses in the bachelor program of most universities. See http://websites.math.leidenuniv.nl/algebra for the course notes Algebra 1, 2, 3 used in Leiden and Delft, or chapter I-VI of S. Lang's 'Algebra', Springer GTM 211. In particular, we assume that the student is familiar with the following notions: group, commutative ring, field, homomorphism, ideal, quotient ring, polynomial ring, domain, fraction field, principal ideal domain, unique factorization domain, finite, algebraic, separable and normal field extension, Galois theory.

Rules about Homework/Exam

The final grade will be a weighted average of the homework grade (25%) and the final exam (75%). For students who retake the final exam, the exam counts for 100% of the final grade, and homework performance will not be taken into account. Your homework grade is the average of your 10 best homework sets, and your exam grade should be at least 5 out of 10. If you volunteer to present your solutions in class, your average will be computed based on your 9 best homework sets.

Lecture notes/Literature

The course notes "Number rings", which include the exercises from which the homework will be taken, are among those available at the Leiden algebra website http://websites.math.leidenuniv.nl/algebra.

 

Lecturers
Eugenia Rosu, Peter Stevenhagen, and Jan Vonk