## ABSTRACT

An important property shared by algebraic number fields and algebraic function fields in one variable is the product formula and it suggests the following definition: a global field is a field with a family of absolute values satisfying the product formula. This chapter finds that every global field is a finite algebraic extension of either Q or a rational function field k(x), where k is a finite field. A small trick allows one to extend this result to the case of a perfect ground field k. By an algebraic number one understands any solution of a polynomial equation with rational coefficients. The chapter shows that its kernel is a finitely generated abelian group (Dirichlet unit theorem) and its cokernel, the divisor class group, is finite. The chapter begins with a look at quadratic extensions, to set the scene and illustrate the concepts introduced so far.