VALUES OF RATIONAL FUNCTIONS ON NON-HILBERTIAN FIELDS AND A QUESTION OF WEISSAUER

We answer in the negative a question raised by Fried and Jarden, askin g whether the quotient field of a unique factorization domain with inf initely many primes is necessarily hilbertian. This implies a negative answer to a related question of Weissauer. Our constructions are simp le and take place inside the field of algebraic numbers. Simultaneousl y we investigate the relation of hilbertianity of a field K with the s tructure of the value sets of rational functions on K: we construct a non-hilbertian subfield K of (Q) over bar such that, given any f(1),.. ., f(h) is an element of K(x), each of degree greater than or equal to 2, the union U-i(h)= f(i)(K) does not contain K.