Undecidability and definability for the theory of global fields
HTML articles powered by AMS MathViewer
- by R. S. Rumely PDF
- Trans. Amer. Math. Soc. 262 (1980), 195-217 Request permission
Abstract:
We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse’s Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly defined in all global fields, as well as Gödel functions encoding finite sequences of elements.References
- Emil Artin, Algebraic numbers and algebraic functions, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR 0237460
- E. Artin and J. Tate, Class field theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0223335
- J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
- Ju. L. Eršov, The undecidability of certain fields, Dokl. Akad. Nauk SSSR 161 (1965), 27–29 (Russian). MR 0175785
- Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
- O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Band 117, Springer-Verlag, New York-Heidelberg, 1971. Second printing, corrected. MR 0347768
- Ju. G. Penzin, Undecidability of fields of rational functions over fields of characteristic $2$, Algebra i Logika 12 (1973), 205–210, 244 (Russian). MR 0389875
- Julia Robinson, On the decision problem for algebraic rings, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 297–304. MR 0146083
- Julia Robinson, Definability and decision problems in arithmetic, J. Symbolic Logic 14 (1949), 98–114. MR 31446, DOI 10.2307/2266510
- Julia Robinson, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959), 950–957. MR 112842, DOI 10.1090/S0002-9939-1959-0112842-7
- Raphael M. Robinson, Undecidable rings, Trans. Amer. Math. Soc. 70 (1951), 137–159. MR 41081, DOI 10.1090/S0002-9947-1951-0041081-0
- Raphael M. Robinson, Arithmetical definability of field elements, J. Symbolic Logic 16 (1951), 125–126. MR 42357, DOI 10.2307/2266685
- Raphael M. Robinson, The undecidability of pure transcendental extensions of real fields, Z. Math. Logik Grundlagen Math. 10 (1964), 275–282. MR 172803, DOI 10.1002/malq.19640101803
- Joseph R. Shoenfield, Mathematical logic, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR 0225631
- André Weil, Basic number theory, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag, New York-Berlin, 1974. MR 0427267
- Carl Siegel, Approximation algebraischer Zahlen, Math. Z. 10 (1921), no. 3-4, 173–213 (German). MR 1544471, DOI 10.1007/BF01211608
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 262 (1980), 195-217
- MSC: Primary 03D35; Secondary 10N05, 12L05
- DOI: https://doi.org/10.1090/S0002-9947-1980-0583852-6
- MathSciNet review: 583852