Algebraic Number Theory

Introduction to algebraic number theory This course (36 hours) is a relatively elementary course which requires minimal prerequisites from commutative algebra for its understanding. Its first part (modules over principal ideal domains, Noetherian modules) follows to a certain extent the book of P. Samuel "Algebraic Theory of Numbers". Then integrality over rings, algebraic extensions of fields, field isomorphisms, norms and traces are discussed in the second part. In the main third part Dedekind rings, factorization in Dedekind rings, norms of ideals, splitting of prime ideals in field extensions, finiteness of the ideal class group and Dirichlet's theorem on units are treated. The exposition sometimes uses methods of presentation from the book of D. A. Marcus "Number Fields".