Fundamental Properties of Ordered Fields

Abstract

We discuss fundamental algebraic-order-topological properties of ordered fields. In fact, the absolute value in any ordered field has properties similar to those of real numbers. We give a simple proof of the equivalence between the Archimedean property and the density of the rational subfield. We also provide equivalent conditions for an ordered field to be Archimedean, involving convergence of certain sequences and the geometric series test.