Entry requirements

120 credits including Algebraic Structures. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of 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.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.