On the structure of hyperfields obtained as quotients of fields
HTML articles powered by AMS MathViewer
- by Matthew Baker and Tong Jin
- Proc. Amer. Math. Soc. 149 (2021), 63-70
- DOI: https://doi.org/10.1090/proc/15207
- Published electronically: October 20, 2020
- PDF | Request permission
Abstract:
We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients of fields. The main ingredients in the proof are the Weil bounds from number theory and a result from Ramsey theory.References
- Vitaly Bergelson and Daniel B. Shapiro, Multiplicative subgroups of finite index in a ring, Proc. Amer. Math. Soc. 116 (1992), no. 4, 885–896. MR 1095220, DOI 10.1090/S0002-9939-1992-1095220-5
- Henri Cohen, Number theory. Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, Springer, New York, 2007. MR 2312337
- H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1935), 151–182 (German). MR 1581445, DOI 10.1515/crll.1935.172.151
- A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229. MR 143712, DOI 10.1090/S0002-9947-1963-0143712-1
- 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
- Marc Krasner, A class of hyperrings and hyperfields, Internat. J. Math. Math. Sci. 6 (1983), no. 2, 307–311. MR 701303, DOI 10.1155/S0161171283000265
- M. Marshall, Real reduced multirings and multifields, J. Pure Appl. Algebra 205 (2006), no. 2, 452–468. MR 2203627, DOI 10.1016/j.jpaa.2005.07.011
- Ch. G. Massouros, Methods of constructing hyperfields, Internat. J. Math. Math. Sci. 8 (1985), no. 4, 725–728. MR 821630, DOI 10.1155/S0161171285000813
- Gerhard Turnwald, Multiplicative subgroups of finite index in a division ring, Proc. Amer. Math. Soc. 120 (1994), no. 2, 377–381. MR 1215206, DOI 10.1090/S0002-9939-1994-1215206-9
Bibliographic Information
- Matthew Baker
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- MR Author ID: 638188
- Email: mbaker@math.gatech.edu
- Tong Jin
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- Address at time of publication: Taishan College, Shandong University, Jinan, People’s Republic of China
- Email: tjinmath@outlook.com
- Received by editor(s): February 14, 2020
- Received by editor(s) in revised form: April 29, 2020
- Published electronically: October 20, 2020
- Communicated by: Romyar T. Sharifi
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 63-70
- MSC (2010): Primary 12K99; Secondary 11T30
- DOI: https://doi.org/10.1090/proc/15207
- MathSciNet review: 4172586