Multiplicative subgroups of finite index in a ring
HTML articles powered by AMS MathViewer
- by Vitaly Bergelson and Daniel B. Shapiro PDF
- Proc. Amer. Math. Soc. 116 (1992), 885-896 Request permission
Abstract:
If $G$ is a subgroup of finite index in the multiplicative group of an infinite field $K$ then $G - G = K$. Similar results hold for various rings.References
- Eberhard Becker, Extended Artin-Schreier theory of fields, Rocky Mountain J. Math. 14 (1984), no. 4, 881–897. Ordered fields and real algebraic geometry (Boulder, Colo., 1983). MR 773127, DOI 10.1216/RMJ-1984-14-4-881
- Vitaly Bergelson, Ergodic Ramsey theory, Logic and combinatorics (Arcata, Calif., 1985) Contemp. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1987, pp. 63–87. MR 891243, DOI 10.1090/conm/065/891243
- Pedro Berrizbeitia, Additive properties of multiplicative subgroups of finite index in fields, Proc. Amer. Math. Soc. 112 (1991), no. 2, 365–369. MR 1057940, DOI 10.1090/S0002-9939-1991-1057940-7
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
- Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1980. MR 591457
- Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549 A. J. Hahn and O. T. O’Meara, The classical groups and $K$-theory, Springer-Verlag, Berlin, 1989. E. Hewitt and K. A. Ross, Abstract harmonic analysis, vol. 1, Springer-Verlag, Berlin, 1963.
- Nathan Jacobson, Basic algebra. I, W. H. Freeman and Co., San Francisco, Calif., 1974. MR 0356989
- David B. Leep and Daniel B. Shapiro, Multiplicative subgroups of index three in a field, Proc. Amer. Math. Soc. 105 (1989), no. 4, 802–807. MR 963572, DOI 10.1090/S0002-9939-1989-0963572-X
- Stan Wagon, The Banach-Tarski paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge, 1985. With a foreword by Jan Mycielski. MR 803509, DOI 10.1017/CBO9780511609596
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 885-896
- MSC: Primary 16B99; Secondary 05D10, 12E99
- DOI: https://doi.org/10.1090/S0002-9939-1992-1095220-5
- MathSciNet review: 1095220