DOI QR코드

DOI QR Code

ON GENERALIZED GRADED CROSSED PRODUCTS AND KUMMER SUBFIELDS OF SIMPLE ALGEBRAS

  • Bennis, Driss (Mohammed V University in Rabat Faculty of Sciences Research Center CeReMAR) ;
  • Mounirh, Karim (Mohammed V University in Rabat Faculty of Sciences Research Center CeReMAR) ;
  • Taraza, Fouad (Mohammed V University in Rabat Faculty of Sciences Research Center CeReMAR)
  • 투고 : 2018.08.01
  • 심사 : 2019.02.08
  • 발행 : 2019.07.31

초록

Using generalized graded crossed products, we give necessary and sufficient conditions for a simple algebra over a Henselian valued field (under some hypotheses) to have Kummer subfields. This study generalizes some known works. We also study many properties of generalized graded crossed products and conditions for embedding a graded simple algebra into a matrix algebra of a graded division ring.

키워드

참고문헌

  1. M. Boulagouaz, The graded and tame extensions, in Commutative ring theory (Fes, 1992), 27-40, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.
  2. M. Boulagouaz, Algebre a division graduee centrale, Comm. Algebra 26 (1998), no. 9, 2933-2947. https://doi.org/10.1080/00927879808826318
  3. B. Fein, D. J. Saltman, and M. Schacher, Minimal embeddings of central simple algebras, J. Algebra 133 (1990), no. 2, 404-423. https://doi.org/10.1016/0021-8693(90)90277-U
  4. Y.-S. Hwang and A. R. Wadsworth, Algebraic extensions of graded and valued fields, Comm. Algebra 27 (1999), no. 2, 821-840. https://doi.org/10.1080/00927879908826464
  5. Y.-S. Hwang and A. R. Wadsworth, Correspondences between valued division algebras and graded division algebras, J. Algebra 220 (1999), no. 1, 73-114. https://doi.org/10.1006/jabr.1999.7903
  6. J. Minac and A. R. Wadsworth, The u-invariant for algebraic extensions, in K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 333-358, Proc. Sympos. Pure Math., 58, Part 2, Amer. Math. Soc., Providence, RI, 1995.
  7. P. J. Morandi and B. A. Sethuraman, Kummer subfields of tame division algebras, J. Algebra 172 (1995), no. 2, 554-583. https://doi.org/10.1016/S0021-8693(05)80015-8
  8. K. Mounirh, Kummer subfields of tame division algebras over Henselian fields, J. Pure Appl. Algebra 214 (2010), no. 4, 440-448. https://doi.org/10.1016/j.jpaa.2009.06.013
  9. K. Mounirh, Discriminants of orthogonal involutions on central simple algebras with tame gauges, J. Pure Appl. Algebra 215 (2011), no. 11, 2547-2558. https://doi.org/10.1016/j.jpaa.2011.02.004
  10. K. Mounirh and A. R. Wadsworth, Subfields of nondegenerate tame semiramified division algebras, Comm. Algebra 39 (2011), no. 2, 462-485. https://doi.org/10.1080/00927871003591926
  11. J.-F. Renard, J.-P. Tignol, and A. R. Wadsworth, Graded Hermitian forms and Springer's theorem, Indag. Math. (N.S.) 18 (2007), no. 1, 97-134. https://doi.org/10.1016/S0019-3577(07)80010-3
  12. J.-P. Tignol and S. A. Amitsur, Kummer subfields of Malcev-Neumann division algebras, Israel J. Math. 50 (1985), no. 1-2, 114-144. https://doi.org/10.1007/BF02761120
  13. J.-P. Tignol and A. R. Wadsworth, Value functions and associated graded rings for semisimple algebras, Trans. Amer. Math. Soc. 362 (2010), no. 2, 687-726. https://doi.org/10.1090/S0002-9947-09-04681-9
  14. J.-P. Tignol and A. R. Wadsworth, Value functions on simple algebras, and associated graded rings, Springer Monographs in Mathematics, Springer, Cham, 2015.