Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Totally ramified valuations on finite-dimensional division algebras


Authors: J.-P. Tignol and A. R. Wadsworth
Journal: Trans. Amer. Math. Soc. 302 (1987), 223-250
MSC: Primary 16A39; Secondary 12E15, 16A10
DOI: https://doi.org/10.1090/S0002-9947-1987-0887507-6
MathSciNet review: 887507
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Division algebras $ D$ with valuation $ v$ are studied, where $ D$ is finite-dimensional and totally ramified over its center $ F$ (i.e., the ramification index of $ v$ over $ v{\vert _F}$ equals $ [D:F]$). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When $ v{\vert _F}$ is Henselian a full classification of the $ F$-subalgebras of $ D$ is given. When $ F$ has a Henselian valuation $ v$ with separably closed residue field and $ A$ is any tame central simple $ F$-algebra, an algorithm is given for computing the underlying division algebra of $ A$ from a suitable subgroup of $ {A^{\ast}}/{F^{\ast}}$. Some examples are constructed using this valuation theory, including the first example of finite-dimensional $ F$-central division algebras $ {D_1}$ and $ {D_2}$ with $ {D_1}{ \otimes _F}{D_2}$ not a division ring, but $ {D_1}$ and $ {D_2}$ having no common subfield $ K \supsetneqq F$.


References [Enhancements On Off] (What's this?)

  • [A$ _{1}$] A. A. Albert, On the Wedderburn norm condition for cyclic algebras, Bull. Amer. Math. Soc. 37 (1931), 301-312. MR 1562142
  • [A$ _{2}$] -, Noncyclic algebras of degree and exponent four, Trans. Amer. Math. Soc. 35 (1933), 112-121. MR 1501674
  • [A$ _{3}$] -, Structure of algebras, Amer. Math. Soc. Colloq. Publ., vol. 24, Amer. Math. Soc., Providence, R. I., 1961.
  • [Am] S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408-420. MR 0318216 (47:6763)
  • [AS] S. A. Amitsur and D. Saltman, Generic abelian crossed products and $ p$-algebras, J. Algebra 51 (1978), 76-87. MR 0491789 (58:10988)
  • [B] N. Bourbaki, Algèbre commutative, Chapitre VI, Valuations, Hermann, Paris, 1961.
  • [C] P. M. Cohn, On extending valuations in division algebras, Studia Sci. Math. Hungar. 16 (1981), 65-70. MR 703642 (84f:16025)
  • [CM] P. M. Cohn and M. Mahdavi-Hezavehi, Extensions of valuations on skew fields, Ring Theory, Antwerp 1980 (F. Van Oystaeyen, ed.), Lecture Notes in Math., vol. 825, Springer-Verlag, Berlin and New York, 1980, pp. 28-41. MR 590783 (81m:12024)
  • [D$ _{1}$] P. Draxl, Skew fields, London Math. Soc. Lecture Note Series, no. 81, Cambridge Univ. Press, Cambridge, 1983. MR 696937 (85a:16022)
  • [D$ _{2}$] -, Ostrowski's theorem for Henselian valued skew fields, J. Reine Angew. Math. 354 (1984), 213-218. MR 767581 (86g:12008)
  • [DK] P. Draxl and M. Kneser (eds.), $ S{K_1}$ von Schiefkörpern, Lecture Notes in Math., vol. 778, Springer-Verlag, Berlin and New York, 1980. MR 591206 (82b:16014)
  • [E] O. Endler, Valuation theory, Springer-Verlag, New York, 1972. MR 0357379 (50:9847)
  • [H] H. Hasse, Über $ p$-adische Schiefkörper und ihre Bedeutung für die Arithmetik hyperkomplexer Zahlsysteme, Math. Ann. 104 (1931), 495-534. MR 1512683
  • [JW] B. Jacob and A. Wadsworth, A new construction of noncrossed product algebras, Trans. Amer. Math. Soc. 293 (1986), 693-721. MR 816320 (87g:16027)
  • [J$ _{1}$] N. Jacobson, Lectures in abstract algebra, Vol. III, Van Nostrand, Princeton, N.J., 1964. MR 0172871 (30:3087)
  • [J$ _{2}$] -, P. I.-algebras, an introduction, Lecture Notes in Math., vol. 441, Springer-Verlag, Berlin and New York, 1975. MR 0369421 (51:5654)
  • [M] J. Milnor, Introduction to algebraic $ K$-theory, Ann. of Math. Studies, No. 72, Princeton Univ. Press, Princeton, N. J., 1971. MR 0349811 (50:2304)
  • [P] V. P. Platonov, The Tannaka-Artin problem and reduced $ K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 227-261; English transl. in Math. USSR Izv. 10 (1976), 211-243. MR 0407082 (53:10865)
  • [PY] V. P. Platonov and V. I. Yanchevskii, Dieudonné's conjecture on the structure of unitary groups, and division rings over Henselian fields, Dokl. Akad. Nauk SSSR 279 (1984), 546-549; English transl. in Soviet Math. Dokl. 30 (1984), 693-696. MR 771746 (86c:11026)
  • [Ri] P. Ribenboim, Théorie des valuations, Presses Univ. Montréal, Montréal, 1968. MR 0249425 (40:2670)
  • [Ro] J. Rotman, Introduction to the theory of groups, 3rd ed., Allyn and Bacon, Boston, Mass., 1984. MR 745804 (85f:20001)
  • [Sa] D. Saltman, Indecomposable division algebras, Comm. Algebra 7 (1979), 791-817. MR 529494 (80g:16021)
  • [Sch] W. Scharlau, Über die Brauer-Gruppe eines Henselkörpers, Abh. Math. Sem. Univ. Hamburg 33 (1969), 243-249. MR 0255524 (41:185)
  • [S] O. F. G. Schilling, The theory of valuations, Math. Surveys, no. 4, Amer. Math. Soc., Providence, R. I., 1950. MR 0043776 (13:315b)
  • [Ta] J. Tate, Relations between $ {K_2}$ and Galois cohomology, Invent. Math. 36 (1976), 257-274. MR 0429837 (55:2847)
  • [T$ _{1}$] J.-P. Tignol, Sur les décompositions des algèbres à division en produit tensoriel d'algèbres cycliques, Brauer Groups in Ring Theory and Algebraic Geometry (F. Van Oystaeyen and A. Verschoren, eds.) Lecture Notes in Math., vol. 917, Springer-Verlag, Berlin and New York, 1982, pp. 126-145. MR 657427 (83i:16020)
  • [T$ _{2}$] -, On the length of decompositions of central simple algebras in tensor products of symbols, Methods in Ring Theory (F. Van Oystaeyen, ed.), NATO ASI Series, Series C, Vol. 129, Reidel, Dordrecht, 1984, pp. 505-516. MR 770614 (86d:16029)
  • [TA$ _{1}$] J.-P. Tignol and S. A. Amitsur, Totally ramified splitting fields of central simple algebras over Henselian fields, J. Algebra 98 (1986), 95-101. MR 825137 (87d:12005)
  • [TA$ _{2}$] -, Kummer subfields of Malcev-Neumann division algebras, Israel J. Math. 50 (1985), 114-144. MR 788071 (86m:16007)
  • [TA$ _{3}$] -, Symplectic modules, Israel J. Math. 54 (1986), 266-290. MR 853452 (87m:16042)
  • [VG] J. Van Geel, Places and valuations in noncommutative ring theory, Lecture Notes in Pure and Appl. Math., no. 71, Marcel Dekker, New York, 1981. MR 639305 (83d:16021)
  • [W$ _{1}$] A. Wadsworth, $ p$-Henselian fields: $ K$-theory, Galois cohomology, and Witt rings, Pacific J. Math. 105 (1983), 473-495. MR 691616 (84m:12026)
  • [W$ _{2}$] -, Extending valuations to finite-dimensional division algebras, Proc. Amer. Math. Soc. 98 (1986), 20-22. MR 848866 (87i:16025)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A39, 12E15, 16A10

Retrieve articles in all journals with MSC: 16A39, 12E15, 16A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0887507-6
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society