MATT INSALL Hyperalgebraic Primitive Elements for Relational Algebraic and Topological Algebraic Models Abstract. Using nonstandard methods, we generalize the notion of an algebraic prim- itive element to that of an hyperalgebraic primitive element, and show that under mild restrictions, such elements can be found infinitesimally close to any given element of a topological field. Key words: nonstandard methods, topological field, hyperalgebraic elements, infinitesimal, topological model. 1991 Mathematics Subject Classification. 12F05, 12J99, 12L15, 03C20, 03H05. Introduction Fried and Jgrden [8] applied nonstandard methods to Hilbertian fields. Ax and Kochen [2, 3, 4] used ultraproducts to study Diophantine problems over local fields. Robinson [23] suggested that model theory should expand in the direction of topological models. More recently, much work in applications of model theory to algebra involves definability in algebraic structures with some compatible relation on them, e.g. ordered rings and ordered fields. We consider algebraicity, rather than definability, in relational algebraic struc- tures in which a "primitive element theorem" holds. Nonstandard methods are applicable, then, both in a purely algebraic and in a topological algebraic sense. We then add a compatible topology to the original structure. We assume a reasonable level of familiarity with topological field the- ory and elementary nonstandard analysis. For the standard theory, nota- tion is taken primarily from Fraleigh [7], Burris and Sankapanavar [6], and Wi~staw [27]; for nonstandard methods, see Hurd and Loeb [13], and Albev- erio, et. al. [1]. Other sources for the nonstandard methods include [12] and [21]. The purely algebraic ideas and results we present expand those origi- nated in a part of the author's master's thesis, but the results involving topological algebraic structures, relational algebraic structures, and topolog- ical relational algebraic structures, and extensions of automorphisms seem Presented by Robert Goldblatt; Received May 6, 1994; Revised November 30, 1995 Studia Logica 57: 409-418, 1996. 9 1996 Kluwer Academic Publishers. Printed in the Netherlands.