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Israel Journal of Mathematics

, Volume 105, Issue 1, pp 323–335 | Cite as

Values of rational functions on non-hilbertian fields and a question of Weissauer

  • P. Corvaja
  • U. Zannier
Article
  • 53 Downloads

Abstract

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 onK: we construct a non-hilbertian subfieldK of\(\bar {\mathbb{Q}}\) such that, given anyf 1 ,…,f h ∈K(x), each of degree ≥2, the union ∪ z=1 h f z(K) does not containK.

Keywords

Rational Function Prime Ideal Galois Group Number Field Algebraic Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [A] E. Artin,Algebraic Numbers and Algebraic Functions, Gordon & Breach, London, 1965.Google Scholar
  2. [CF] J. W. S. Cassels and A. Fröhlich,Algebraic Number Theory, Academic Press, New York, 1990.Google Scholar
  3. [DZ] P. Dèbes and U. Zannier,Universal Hilbert subsets, Mathematical Proceedings of the Cambridge Philosophical Society, to appear.Google Scholar
  4. [Fr1] M. Fried,On the Sprindzuk-Weissauer approach to universal Hilbert subsets, Israel Journal of Mathematics51 (1985), 347–363.MATHCrossRefMathSciNetGoogle Scholar
  5. [Fr2] M. Fried,Arithmetical properties of value sets of polynomials, Acta Arithmetica15 (1969), 91–115.MATHMathSciNetGoogle Scholar
  6. [Fr3] M. Fried,On Hilbert's irreducibility theorem, Journal of Number Theory6 (1974), 211–231.MATHCrossRefMathSciNetGoogle Scholar
  7. [FrJ] M. Fried and M. Jarden,Field Arithmetic, Springer-Verlag, Berlin, 1986.MATHGoogle Scholar
  8. [Ha] D. Harbater,Galois coverings of the arithmetic line, inNumber Theory — New York, 1984–85 (D. V. and G. V. Chudnovsky, eds.), Lecture Notes in Mathematics1240, Springer-Verlag, Berlin, 1987, pp. 165–195.Google Scholar
  9. [L1] S. Lang,Diophantine Geometry, Interscience Publishers, New York, 1962.MATHGoogle Scholar
  10. [L2] S. Lang,Fundamentals of Diophantine Geometry, Springer-Verlag, Berlin, 1983.MATHGoogle Scholar
  11. [Sch] A. Schinzel,Selected Topics on Polynomials, The University of Michigan Press, Ann Arbor, 1983.Google Scholar
  12. [Se1] J. P. Serre,Lectures on the Mordell-Weil Theorem, Vieweg, 2nd ed., Braunschweig, 1990.Google Scholar
  13. [Se2] J. P. Serre,Topics in Galois Theory, Jones and Bartlett, Boston, 1992.MATHGoogle Scholar
  14. [Ws] R. Weissauer,Der Hilbertsche Irreduzibilitätssatz, Journal für die reine und angewandte Mathematik334 (1982), 203–220.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University 1998

Authors and Affiliations

  1. 1.Dip. Mat. Inf.Università di UdineUdineItaly
  2. 2.Ist. Univ. Arch. D.C.A.VeneziaItaly

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