On the Sprindžuk-Weissauer approach to universal Hilbert subsets

The works of both Sprindžuk (1979–80) and Weissauer (1980) consider the relation between Hilbert subsets of Q and sets consisting of powers of primes. A comparison of their results leads to generalizations and new proofs devoid of eitherp-adic diophantine approximation or of nonstandard arithmetic (§3 and §4). Results of Weissauer, giving new Hibertian infinite extensions of every Hilbertian field, receive short direct standard proofs, and a negative answer is given to a question of Roquette on the relation between Hilbert sets and value sets.