An algebraic number field is a finite extension of the field of rational numbers. In algebraic number fields the familiar unique factorization into prime numbers may not hold, so the notion of a prime ideal is introduced. In Dedekind rings every ideal can be written umquely as a product of prime ideals. It is shown that the set of algebraic integers in an algebraic number field is a Dedekind ring. One studies the decomposition of prime ideals in finite extension fields.
Valuation theory is introduced along with the relation between prime ideals and discrete valuations.
Ramification groups, the different, and the discriminant of an extension are studied. The familiar Chinese remainder theorem reappears in the form of various approximation theorems. Often the adele ring and idele group are studied at this point. Dirichlet's unit theorem and the finiteness of the class number are proved for number fields. Cyclotomic fields and Kummer extensions provide important examples of the general theory.