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The valuation theory of meromorphic function fields over open Riemann surfaces

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References

  1. N. L. Alling, A characterization of Abelianη α-groups in terms of their natural valuation.Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 711–713.

    MATH  MathSciNet  Google Scholar 

  2. —, On the existence of real-closed fields that areη α-sets of power ℵα.Trans. Amer. Math. Soc., 103 (1962) 341–352.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. L. Alling, An application of valuation theory to rings of continuous real and complex-valued functions.Trans. Amer. Math. Soc. (To appear).

  4. B. Banaschewski, Zur Idealtheorie der ganzen Funktionen.Math. Nachr., 19 (1958), 136–160.

    MATH  MathSciNet  Google Scholar 

  5. C. Chevelley,Introduction to the Theory of Algebraic Functions of one Variable, Mathematical Surveys VI, American Mathematical Society, New York, 1951.

    Google Scholar 

  6. H. Florack, Reguläre und meromorphe Funktionen auf nicht geschlossenen Riemannschen Flächen.Schr. Math. Inst. Univ. Münster, 1 (1948).

  7. L. Gillman &M. Jerison,Rings of Continuous Functions. van Nostrand, Princeton, 1960.

    MATH  Google Scholar 

  8. F. Hausdorff,Grundzüge der Mengenlehre, Verlag von Veit & Co., Leipzig, 1914.

    MATH  Google Scholar 

  9. O. Helmer, Divisibility properties of integral functions.Duke Math. J., 6 (1940), 345–356.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Henriksen, On the ideal structure of the ring of entire functions.Pacific. J. Math., 2 (1952), 179–184.

    MATH  MathSciNet  Google Scholar 

  11. —, On the prime ideals of the ring of entire functions.Pacific J. Math., 3 (1953), 711–720.

    MATH  MathSciNet  Google Scholar 

  12. E. Hewitt, Rings of real valued continuous functions, I.Trans. Amer. Math. Soc., 64 (1948), 45–99.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Kakutani,Rings of Analytic Functions. Lectures on Functions of a Complex Variable.W. Kaplan et. al., Univ. of Mich., 1955.

  14. J. L. Kelley,General Topology, van Nostrand, Princeton, 1955.

    MATH  Google Scholar 

  15. S. Kochen, Ultraproducts in the theory of models.Ann. of Math., (2) 74 (1961), 221–261.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Lang,Introduction to Algeraic, Geometry, Interscience Tracts in Pure and Applied Mathematics No. 5, Interscience Publishers, New York, 1958.

    Google Scholar 

  17. R. Narasimhan, Imbedding of open Riemann surfaces.Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., II 7 (1960), 159–165.

    MathSciNet  Google Scholar 

  18. H. L. Royden, Rings of analytic and meromorphic functions,Trans. Amer. Math. Soc., 83 (1956), 269–276.

    Article  MATH  MathSciNet  Google Scholar 

  19. O. F. G. Schilling,The Theory of Valuations. Mathematical Surveys IV. American Mathematical Society, New York, 1950.

    MATH  Google Scholar 

  20. —, Ideal theory on open Riemann surfaces.Bull. Amer. Math. Soc., 52 (1946), 945–963.

    Article  MATH  MathSciNet  Google Scholar 

  21. O. Zariski &P. Samuel,Commutative Algebra, vol. I. van Nostrand, Princeton, 1958.

    Google Scholar 

  22. —,Commutative Algebra, vol. II. van Nostrand, Princeton, 1960.

    MATH  Google Scholar 

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These researches were done, in part, while the author was a N.S.F. post-doctoral fellow at Harvard University.

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Alling, N.L. The valuation theory of meromorphic function fields over open Riemann surfaces. Acta Math. 110, 79–96 (1963). https://doi.org/10.1007/BF02391855

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