Pseudo Algebraically Closed Fields Over Rational Function Fields

Abstract

The following theorem is proved: Let T be an uncountable set of algebraically independent elements over a field Ko. Then K = KO(T) is a Hilbertian field but the set of a E G( K) for which K(a) is PAC is nonmeasurable. Introduction. A field M is said to be pseudo algebraically closed (= PAC) if every nonempty absolutely irreducible variety V defined over M has an M-rational point. If M is an algebraic extension of a field K and every absolutely irreducible polynomial f E K [ X, Y], separable in Y, has infinitely many M-rational zeros, then M is PAC. This is a combination of Ax's application of descent [1] and the generic hyperplane intersection method as in Frey [3]. If a,,...,ae are e-elements of the absolute Galois group G(K) of K, then K(a) denotes the fixed field in K of a1,. . . ,0e. Here K is the algebraic closure of K. We denote by 4 the normalized Haar measure of G(K )e. It is proved in [6, Lemma 2.4] that if K is a Hilbertian field, if f E K[X, Y] is an absolutely irreducible polynomial and if A(f) = (a E G(K)e If has a K(a)-zero), then u(A(f )) = 1. If in addition K is countable, then there are only countably many f 's and therefore the intersection of all the A( f )'s is also a set of measure 1. Thus the set Se(K) = (a E G(K )e I K(a) is PAC} has measure 1. This basic result, which is called the Nullstellensatz in [6], has been the cornerstone for several model theoretic investigations of the fields K( a) (cf. [9, 7 and 4]). If K is uncountable, then the above argument is not valid any more. It is our aim in this note to show that indeed the Nullstellensatz itself is not true in this case. More precisely, we prove THEOREM. Let T be an uncountable set of algebraically independent elements over a field Ko. Then K = KO T) is a Hilbertian field but Se( K) is a nonmeasurable subset of G( K )e for every positive integer e. 1. The Haar measure of a profinite group. Let G be a profinite group and consider the boolean algebra of open-closed sets in G. They are finite unions of left cosets xN, where N are open normal subgroups. The a-algebra generated by the open-closed sets is denoted by 6J0. Every open subset of G is a union of open-closed sets. We Received by the editors Februarv 5. 1982 and, in revised form, May 27, 1982. 1980 Mathematics Subject Clavsificcation. Primary 1 2F20. Kev words and phrases. PAC fields, Hilbertian fields, Haar measure. ' Partially supported by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities. i) 1983 American Mathematical Society 0002-9939/82/0000-0592/$02. 50