Algebraic Number Theory
Syllabus, Master's level, 1MA207
This course has been discontinued.
- Code
- 1MA207
- Education cycle
- Second cycle
- Main field(s) of study and in-depth level
- Mathematics A1N
- Grading system
- Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
- Finalised by
- The Faculty Board of Science and Technology, 24 April 2013
- Responsible department
- Department of Mathematics
Entry requirements
120 credits including Algebraic Structures.
Learning outcomes
In order to pass the course the student should be able to
- define the key notions of algebraic number theory and outline their interrelation;
- calculate the most important number theoretical quantities introduced during the course;
- give an account of the fundamental theorems of the course and apply them in specific cases;
- outline important parts of the theory presented during the course, such as the deduction of the four-squares theorem from Minkowski's theorem and Kummer's proof of Fermat's great theorem for regular prime exponents;
- explain the concept of "geometry of numbers" according to Minkowski.
Content
Integral elements in commutative ring extensions. Dedekind rings, their ideal theory and ideal classgroup. Fundamental theorem of Dedekind rings. Quadratic number fields. Cyclotomic fields and cyclotomic integers. Regular and irregular primes. Kummer's lemma. Kummer's proof of Fermat's great theorem for regular prime exponents. Norm and trace of an algebraic number. Ramification index and residue class degree of a prime ideal with respect to an algebraic field extension. Discriminant of an algebraic number field. Lattices in Rn and their quotient torus. Minkowski's theorem. Four-squares theorem. Minkowski's bound.
Instruction
Lectures.
Assessment
Written examination at the end of the course and assignments distributed during the course.