Abstract
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.
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