Noncrossed product division algebras with a Baer ordering
HTML articles powered by AMS MathViewer
- by Patrick J. Morandi and B. A. Sethuraman
- Proc. Amer. Math. Soc. 123 (1995), 1995-2003
- DOI: https://doi.org/10.1090/S0002-9939-1995-1246532-6
- PDF | Request permission
Abstract:
Let $n|m$ be positive integers with the same prime factors, such that ${p^3}|n$ for some prime p. We construct a noncrossed product division algebra D with involution $\ast$, of index m and exponent n, such that D possesses a Baer ordering relative to the involution $\ast$. Using similar techniques we construct indecomposable division algebras with involution possessing a Baer ordering.References
- Reinhold Baer, Linear algebra and projective geometry, Academic Press, Inc., New York, N.Y., 1952. MR 0052795
- Thomas C. Craven, Orderings and valuations on ${}^*$-fields, Rocky Mountain J. Math. 19 (1989), no. 3, 629–646. Quadratic forms and real algebraic geometry (Corvallis, OR, 1986). MR 1043236, DOI 10.1216/RMJ-1989-19-3-629
- Thomas C. Craven, Approximation properties for orderings on $*$-fields, Trans. Amer. Math. Soc. 310 (1988), no. 2, 837–850. MR 973179, DOI 10.1090/S0002-9947-1988-0973179-X
- Thomas C. Craven, Places on ${}^*$-fields and the real holomorphy ring, Comm. Algebra 18 (1990), no. 9, 2791–2820. MR 1063343, DOI 10.1080/00927879008824053
- M. Chacron and A. R. Wadsworth, On decomposing $c$-valued division rings, J. Algebra 134 (1990), no. 1, 182–208. MR 1068421, DOI 10.1016/0021-8693(90)90217-C
- P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, Cambridge, 1983. MR 696937, DOI 10.1017/CBO9780511661907
- Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379, DOI 10.1007/978-3-642-65505-0
- Samuel S. Holland Jr., Orderings and square roots in *-fields, J. Algebra 46 (1977), no. 1, 207–219. MR 447200, DOI 10.1016/0021-8693(77)90402-1
- Samuel S. Holland Jr., $^{\ast }$-valuations and ordered $^{\ast }$-fields, Trans. Amer. Math. Soc. 262 (1980), no. 1, 219–243. MR 583853, DOI 10.1090/S0002-9947-1980-0583853-8
- Samuel S. Holland Jr., Baer ordered $\ast$-fields of the first kind, Israel J. Math. 57 (1987), no. 3, 365–374. MR 889984, DOI 10.1007/BF02766220
- Bill Jacob and Adrian R. Wadsworth, A new construction of noncrossed product algebras, Trans. Amer. Math. Soc. 293 (1986), no. 2, 693–721. MR 816320, DOI 10.1090/S0002-9947-1986-0816320-X
- Tsit Yuen Lam, Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 714331, DOI 10.1090/cbms/052 K. H. Leung, Baer orderings and weak $\ast$-orderings on $\ast$-fields, preprint.
- Ka Hin Leung, Strong approximation property for Baer orderings on $*$-fields, J. Algebra 165 (1994), no. 1, 1–22. MR 1272576, DOI 10.1006/jabr.1994.1095
- Patrick J. Morandi and Adrian R. Wadsworth, Baer orderings with noninvariant valuation ring, Israel J. Math. 68 (1989), no. 2, 241–255. MR 1035892, DOI 10.1007/BF02772663
- Patrick J. Morandi and B. A. Sethuraman, Indecomposable division algebras with a Baer ordering, Comm. Algebra 22 (1994), no. 13, 5401–5418. MR 1291037, DOI 10.1080/00927879408825137
- Joseph J. Rotman, The theory of groups. An introduction, 2nd ed., Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, Inc., Boston, Mass., 1973. MR 0442063
- J.-P. Tignol and A. R. Wadsworth, Totally ramified valuations on finite-dimensional division algebras, Trans. Amer. Math. Soc. 302 (1987), no. 1, 223–250. MR 887507, DOI 10.1090/S0002-9947-1987-0887507-6
- Adrian R. Wadsworth, Extending valuations to finite-dimensional division algebras, Proc. Amer. Math. Soc. 98 (1986), no. 1, 20–22. MR 848866, DOI 10.1090/S0002-9939-1986-0848866-8
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 1995-2003
- MSC: Primary 16K20; Secondary 12E15, 16W10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1246532-6
- MathSciNet review: 1246532