Acta Mathematica

The valuation theory of meromorphic function fields over open Riemann surfaces

Norman L. Alling

Full-text: Open access


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

Article information

Acta Math. Volume 110 (1963), 79-96.

Received: 11 March 1963
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

1963 © Almqvist & Wiksells Boktryckeri AB


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

Export citation


  • N. L. Alling, A characterization of Abelian ηα-groups in terms of their natural valuation. Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 711–713.
  • —, On the existence of real-closed fields that are ηα-sets of power ℵα. Trans. Amer. Math. Soc., 103 (1962) 341–352.
  • N. L. Alling, An application of valuation theory to rings of continuous real and complex-valued functions. Trans. Amer. Math. Soc. (To appear).
  • B. Banaschewski, Zur Idealtheorie der ganzen Funktionen. Math. Nachr., 19 (1958), 136–160.
  • C. Chevelley, Introduction to the Theory of Algebraic Functions of one Variable, Mathematical Surveys VI, American Mathematical Society, New York, 1951.
  • H. Florack, Reguläre und meromorphe Funktionen auf nicht geschlossenen Riemannschen Flächen. Schr. Math. Inst. Univ. Münster, 1 (1948).
  • L. Gillman & M. Jerison, Rings of Continuous Functions. van Nostrand, Princeton, 1960.
  • F. Hausdorff, Grundzüge der Mengenlehre, Verlag von Veit & Co., Leipzig, 1914.
  • O. Helmer, Divisibility properties of integral functions. Duke Math. J., 6 (1940), 345–356.
  • M. Henriksen, On the ideal structure of the ring of entire functions. Pacific. J. Math., 2 (1952), 179–184.
  • —, On the prime ideals of the ring of entire functions. Pacific J. Math., 3 (1953), 711–720.
  • E. Hewitt, Rings of real valued continuous functions, I. Trans. Amer. Math. Soc., 64 (1948), 45–99.
  • S. Kakutani, Rings of Analytic Functions. Lectures on Functions of a Complex Variable. W. Kaplanet. al., Univ. of Mich., 1955.
  • J. L. Kelley, General Topology, van Nostrand, Princeton, 1955.
  • S. Kochen, Ultraproducts in the theory of models. Ann. of Math., (2) 74 (1961), 221–261.
  • S. Lang, Introduction to Algeraic, Geometry, Interscience Tracts in Pure and Applied Mathematics No. 5, Interscience Publishers, New York, 1958.
  • R. Narasimhan, Imbedding of open Riemann surfaces. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., II 7 (1960), 159–165.
  • H. L. Royden, Rings of analytic and meromorphic functions, Trans. Amer. Math. Soc., 83 (1956), 269–276.
  • O. F. G. Schilling, The Theory of Valuations. Mathematical Surveys IV. American Mathematical Society, New York, 1950.
  • —, Ideal theory on open Riemann surfaces. Bull. Amer. Math. Soc., 52 (1946), 945–963.
  • O. Zariski & P. Samuel, Commutative Algebra, vol. I. van Nostrand, Princeton, 1958.
  • —, Commutative Algebra, vol. II. van Nostrand, Princeton, 1960.