An algebraic number field is a field obtained by adjoining to the rational numbers the roots of an irreducible rational polynomial. Algebraic number theory is the study of properties of such fields. This course will cover the following topics: number fields, rings of integers, factorization in Dedekind domains, class numbers and class groups, units in rings of integers , valuations and local fields, and if time permits decomposition of primes, and zeta functions of number fields. These tools provide solutions to several problems which are elementary to state but surprisingly difficult to resolve, including Pell's equation, quadratic reciprocity, the two squares theorem and the four squares theorem (every positive integer is a sum of four square integers).
The students should be able to explain the key notions of algebraic number theory and outline their interrelation
The students should be able to summarize the fundamental theorems of the course and apply them in specific cases.
The students should be able to calculate the most important number theoretical quantities introduced during the course
The students should be able to explain the concept of "geometry of numbers" according to Minkowski.