ABSTRACT
In this chapter we present the fundamentals of general theory of formally real fields. These include the part of Artin-Schreier theory needed in quadratic form theory. Thus we discuss real closed fields, orderings and totally positive elements. As an example of application, we introduce the total signature of a quadratic form, depending on all possible orderings of the ground field. This is of great significance for the structure of the Witt ring of a formally real field. We will use these ideas in the next chapter to establish a link between prime ideals of the Witt ring W(K) and the orderings of the field K.
