On the Sprindžuk-Weissauer approach to universal Hilbert subsets (Q1066944)

The works of both, Sprindzhuk and Weissauer, consider the relation between Hilbert subsets of \({\mathbb{Q}}\) and sets consisting of powers of primes. A comparison of their results leads to generalizations and new proofs devoid of either p-adic diophantine approximation or of nonstandard arithmetic. Results of Weissauer, giving new Hilbertian infinite extensions of every Hilbertian field, receive short direct standard proofs, and a negative answer is given to a question, attributed by the author to the reviewer, on the relation between Hilbert sets and value sets. References: \textit{D. Hilbert}, Über die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 110, 104-129 (1892); \textit{V. G. Sprindzhuk}, ibid. 340, 26-52 (1983; Zbl 0497.12001); \textit{R. Weissauer}, ibid. 334, 203-220 (1982; Zbl 0486.12008); \textit{R. Klein}, ibid. 337, 171-194 (1982; Zbl 0477.12029). See also \textit{M. Yasumoto}, Hilbert irreducibility sequences and nonstandard arithmetic (to appear).