Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Fields with few extensions


Authors: J. Knopfmacher and A. M. Sinclair
Journal: Proc. Amer. Math. Soc. 29 (1971), 255-258
MSC: Primary 12.70
DOI: https://doi.org/10.1090/S0002-9939-1971-0274426-5
MathSciNet review: 0274426
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a valued field $ \Lambda $ with only a finite number of nonisomorphic valued extensions is equal to the complex field C or is real closed with $ C = \Lambda (\surd ( - 1))$.


References [Enhancements On Off] (What's this?)

  • [1] R. Baer and H. Hasse, Zusammenhung und Dimension topologischer Körperräume, J. Reine Angew. Math. 167 (1932), 40-45.
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles, No. 1308, Hermann, Paris, 1964 (French). MR 0194450
  • [3] Ervin Fried, Algebraically closed fields as finite extensions, Mat. Lapok 7 (1956), 47–60 (Hungarian, with Russian and English summaries). MR 0099977
  • [4] Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871
  • [5] Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
  • [6] Lawrence Narici, On nonarchimedian Banach algebras, Arch. Math. (Basel) 19 (1968), 428–435. MR 0236711, https://doi.org/10.1007/BF01898426
  • [7] Niel Shilkret, Non-Archimedean Gelfand theory, Pacific J. Math. 32 (1970), 541–550. MR 0257752

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 12.70

Retrieve articles in all journals with MSC: 12.70


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0274426-5
Keywords: Valuations, real closed fields, algebraically closed fields, real field, complex field
Article copyright: © Copyright 1971 American Mathematical Society