Fields of Number Theory

The first two chapters have been done in great generality. Before we can move on to the deeper results of number theory we shall have to make additional assumptions about the underlying field F. We shall do this by explicitly stating our fields of interest. They are the field of rational numbers or any field of rational functions in one variable over a finite field of coefficients, all finite extensions of these fields, and all completions thereof. By restricting ourselves to these fields we obtain two additional properties. Roughly speaking, the first of these properties is one of finiteness of the residue class field and the second is one of dependence among the valuations. These are actually the decisive properties that distinguish the rest of the arithmetic theory from the first two chapters. In fact it is possible to axiomatize these properties1 and to show that they lead directly to the fields of number theory, but we shall not go into that here.