Some New Results On Ordered Fields

Abstract

The author shows there is a non-Archimedean ordering of the field R(x, y), where x and y are algebraically independent over K, for which the identity is the only order-preserving automorphism. The author proves the partly known result that the following statements about an ordered field K are equivalent: (1) each polynomial in K[x] satisfies the intermediate value theorem; (2) if f 2 K[x] and a < b, then f takes on its maximum value at some c 2 [a, b]; (3) K is real closed. A (not necessarily ordered) field K is said to have the extension property if each automorphism of K(x), where x is transcendental over K, is an extension of an automorphism of K. The author gives sufficient conditions for a field to have the the extension property. For example, a field has the extension property if, for some fixed integer n greater than two, each polynomial xn−ax−1, a 2 K, has a root in K.