Defining algebraic elements

Abstract

If T is the theory of fields and A is a subfield of B, an element b in B is algebraic over A if and only if there is a non-zero polynomial p(x), with coefficients from A and p(b) = 0. A. Robinson, B. Jónsson and M. Morley all generalise this notion of an algebraic element to more general theories and structures. Robinson's approach involves using a first-order formula φ(x) over a language L(A) as an analogue to the notion of polynomial. Jónsson considers monomorphic maps between structures in order to produce a definition of algebraic elements for universal theories with the Amalgamation Property. Morley uses types to arrive at his definition. The three definitions are all generalisations of the classical notion. The three approaches are quite different yet P. Bacsich showed that for a universal theory with the Amalgamation Property, the three definitions of algebraic element are equivalent. Jónsson proved a generalised form of the Amalgamation Property and proved, in a new way, that the class of all fields has the Amalgamation property. Algebraic injective hulls and related topics are used in Bacsich's proof.