No Access

ON THE CLASSIFICATION OF CENTRALLY FINITE ALTERNATIVE DIVISION RINGS SATISFYING ALGEBRAIC CLOSURE CONDITIONS

RICCARDO GHILONI

Department of Mathematics, University of Trento, I–38123, Povo-Trento, Italy

https://doi.org/10.1142/S0219498812500880Cited by:0 (Source: Crossref)

Abstract

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

Keywords:

AMSC: 17A35, 12E15, 17D05

References

A. Berarducci and M. Otero, Ann. Pure Appl. Logic 107(3), 87 (2001), DOI: 10.1016/S0168-0072(00)00027-0. Crossref, Web of Science, Google Scholar
J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 36 (Springer-Verlag, Berlin, 1998). Translated from the 1987 French original. Revised by the authors . Google Scholar
R. H. Bruck and E. Kleinfeld, Proc. Amer. Math. Soc. 2(6), 878 (1951), DOI: 10.1090/S0002-9939-1951-0045099-9. Crossref, Web of Science, Google Scholar
H.-D. Ebbinghaus et al. , Numbers , Graduate Texts in Mathematics 123 ( Springer-Verlag , New York , 1990 ) . Crossref, Google Scholar
S. Eilenberg and I. Niven, Bull. Amer. Math. Soc. 50, 246 (1944), DOI: 10.1090/S0002-9904-1944-08125-1. Crossref, Web of Science, Google Scholar
S. Eilenberg and N. Steenrod , Foundations of Algebraic Topology ( Princeton University Press , Princeton, NJ , 1952 ) . Crossref, Google Scholar
G. Gentili and D. Struppa, Milan J. Math. 76, 15 (2008), DOI: 10.1007/s00032-008-0093-0. Crossref, Web of Science, Google Scholar
R. Ghiloni and A. Perotti, Ann. Mat. Pura Appl. 190(3), 539 (2011), DOI: 10.1007/s10231-010-0162-1. Crossref, Web of Science, Google Scholar
R. Ghiloni and A. Perotti, Adv. Math. 226(2), 1662 (2011), DOI: 10.1016/j.aim.2010.08.015. Crossref, Web of Science, Google Scholar
N. Jacobson , Basic Algebra II , 2nd edn. ( W. H. Freeman , 1989 ) . Crossref, Google Scholar
Y.-L. Jou, Acad. Sinica Sci. Record 3, 29 (1950). Google Scholar
T. Y. Lam , A First Course in Noncommutative Rings ( Springer-Verlag , New York , 1991 ) . Crossref, Google Scholar
I. Niven, Amer. Math. Monthly 48, 654 (1941), DOI: 10.2307/2303304. Crossref, Google Scholar
H. Rodríguez-Ordóñez, Expo. Math. 25(4), 355 (2007). Crossref, Web of Science, Google Scholar
R. D. Schafer , An Introduction to Nonassociative Algebras ( Academic Press , New York , 1966 ) . Google Scholar
R. Serôdio, Adv. Appl. Clifford Algebras 17, 245 (2007). Crossref, Web of Science, Google Scholar
M. Shiota , Nash Manifolds , Lecture Notes in Mathematics 1269 ( Springer-Verlag , Berlin , 1987 ) . Crossref, Google Scholar