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Läuchli's algebraic closure of Q

  • Wilfrid Hodges (a1)
Extract

H. Läuchli (9) constructed, within a model of a weak form of set theory, an algebraic closure L of the field Q of rationals which had no real-closed subfield. Läuchli's construction is easily transferred to a model N of ZF (= Zermelo–Fraenkel set theory without the axiom of Choice), and it follows at once that neither of the two following statements is provable from ZF alone:

Every algebraic closure of Q has a real-closed subfield. (1)

There is, up to isomorphism, at most one algebraic closure of Q. (2)

Copyright
COPYRIGHT: © Cambridge Philosophical Society 1976
References
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(1)Cassels, J. W. S. and Fröhlich, A. (eds.). Algebraic number theory (Academic Press: London, 1967).
(2)Hardy, G. H. and Wright, E. M.An introduction to the theory of numbers, 4th edn (Oxford University Press, 1960).
(3)Hodges, Wilfrid. On the effectivity of some field constructions. Proc. London Math. Soc. (to appear).
(4)Hodges, Wilfrid. Six impossible rings, J. Algebra 31 (1974), 218–244.
(5)Jarden, Moshe. Elementary statements over large algebraic fields. Trans. Amer. Math. Soc. 164 (1972), 67–91.
(6)Jech, T. and Sochor, A.Applications of the θ-model. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 351–355.
(7)Kuyk, Willem. Extensions de corps hilbertiens. J. Algebra 14 (1970), 112–124.
(8)Lang, Serge. Algebra (Addison-Wesley; Reading, Mass., 1969).
(9)Läuchli, H.Auswahlaxiom in der Algebra. Comment. Math. Helv. 37 (1962), 1–18.
(10)Neukirch, Jürgen. Klassenkörpertheorie (Bibliographisches Institut; Mannheim, 1969).
(11)Plotkin, Jacob Manuel. Generic embeddmgs. J. Symbolic Logic 34 (1969), 388–394.
(12)Weiss, Edwin. Algebraic number theory (McGraw-Hill; New York, 1963).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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