Algebraic Number Theory [Fall 2019]
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.
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).
B. de Smit (UL) & P. Stevenhagen (UL)