Places of A-fields

Abstract

By an algebraic number-field, it is customary to understand a finite algebraic extension of Q. One main object of this book, and of number-theory in general, is to study algebraic number-fields by means of their embeddings into local fields. In the last century, however, it was discovered that the methods by which this can be done may be applied with very little change to certain fields of characteristic p > 1; and the simultaneous study of these two types of fields throws much additional light on both of them. With this in mind, we introduce as follows the fields which will be considered from now on:

• Definition 1. A field will be called an A-field if it is either a finite algebraic extension of Q or a finitely generated extension of a finite prime field F_p, of degree of transcendency 1 over F_p.