Transcendental extensions of field topologies on countable fields

Abstract

Let be a field topology on a countable field K and let K(x) be a simple transcendental extension of K. Then there exists a field topology 9' for K(x) such that 'l K=Y. Let K be a countable field and let 91 be a fundamental system of neighborhoods at zero. We identify K with the field of constant functions on K and K(x) with the field R(K) of rational functions over K. If D is a nonprincipal filter on K, then we define for each U E X, UD to be the set of all f E R(K) with {rjf(r) E U} E D and 91D to be the set {UDjU E %}. 1D1K=9, and % D is a filter base. But 91D is not necessarily a fundamental system of neighborhoods at zero of a field topology on R(K). 1. DEFINITION. A filter D on K is called 91-generic, if for each U E 91 and for eachf E R(K) there is a V E W such that {rIf(r)Vc U} E D. 2. THEOREM. If D is a 9-genericfilter on K, then %D is a fundamental system on R(K) such that 91D1K=9. PROOF. To see that %D defines a group topology on R(K), let UD C SD be given. Since 91 is a fundamental system there is a V such that VVc U. Suppose f, g E VD. Then {rjf(r) E V} E D and {r|g(r) E V} E D. By {rlf(r)-g(r) e U}{rlf(r) E V} n{r|g(r) E V} we have that f-g E UD . Thus VD_ VDC UD. By a similar argument, it can be seen that inversion and multiplication at zero are continuous. So it remains to show that multiplication is continuous everywhere. Let f cR(K) and UD C91D be given. Since D is 9-generic there is a V such that {rlf(r) Vc U} E D. Suppose g E VD. Then {r|g(r) e V} e D. {rjg(r) f (r) c( U}) {rJf (r) * Vc U})r{rjg(r) e V} and therefore f -g E UD. Thus, multiplication is continuous everywhere. O Received by the editors March 2, 1972 and, in revised form, July 26, 1972. AMS (MOS) subject classifications. Primary 12J99. ? American Mathematical Society 1973