Ordered Residue Class Rings

Abstract

We saw in the preceding chapter that every residue class field of C or C* modulo a fixed maximal ideal is isomorphic with the real field R. The present chapter initiates the study of residue class fields modulo arbitrary maximal ideals. Each such field has the following properties, as will be shown: it is a totally ordered field, whose order is induced by the partial order in C, and the image of the set of constant functions is an isomorphic copy—necessarily order-preserving—of the real field.