We answer in the negative a question raised by Fried and Jarden, asking whether the quotient field of a unique factorization domain with infinitely many primes is necessarily hilbertian.
This implies a negative answer to a related question of Weissauer. Our constructions are simple and take place inside the field of algebraic numbers. Simultaneously we investigate the relation of hilbertianity of a fieldK with the structure of the value sets of rational functions on K: we construct a non-hilbertian subfield K of Q such that, given anyf 1 ,...,f h ∈K(x), each of degree ≥2, the union ∪ z=1 h f z(K) does not contain K.
