Abstract:
A multiplicative theory of finite-dimensional division rings with involutions of the first kind is developed in connection with Dieudonné's conjecture, and the structure of arbitrary division rings over Henselian fields is studied. The unitary Whitehead group UK1(τ,A) is computed for Henselian division rings A with an involution τ. Classes of division rings for which Dieudonné's conjecture has an affirmative answer are exhibited.
Bibliography: 17 titles.
Citation:
V. P. Platonov, V. I. Yanchevskii, “Dieudonné's conjecture on the structure of unitary groups over a division ring, and Hermitian K-theory”, Math. USSR-Izv., 25:3 (1985), 573–599
\Bibitem{PlaYan84}
\by V.~P.~Platonov, V.~I.~Yanchevskii
\paper Dieudonn\'e's conjecture on the structure of unitary groups over a~division ring, and Hermitian $K$-theory
\jour Math. USSR-Izv.
\yr 1985
\vol 25
\issue 3
\pages 573--599
\mathnet{http://mi.mathnet.ru/eng/im1518}
\crossref{https://doi.org/10.1070/IM1985v025n03ABEH001308}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=772116}
\zmath{https://zbmath.org/?q=an:0593.16012|0579.16011}
Linking options:
https://www.mathnet.ru/eng/im1518
https://doi.org/10.1070/IM1985v025n03ABEH001308
https://www.mathnet.ru/eng/im/v48/i6/p1266
This publication is cited in the following 7 articles:
V. I. Yanchevskiǐ, “Henselian division algebras and reduced unitary Whitehead groups for outer forms of anisotropic algebraic groups of the type $A_n$”, Sb. Math., 213:8 (2022), 1096–1156
R Hazrat, “SK1-like Functors for Division Algebras”, Journal of Algebra, 239:2 (2001), 573
S. I. Adian, E. I. Zel'manov, G. A. Margulis, S. P. Novikov, A. S. Rapinchuk, L. D. Faddeev, V. I. Yanchevskii, “Vladimir Petrovich Platonov (on his 60th birthday)”, Russian Math. Surveys, 55:3 (2000), 601–610
M. Chacron, “Decomposing and ordering a certain crossed product”, Communications in Algebra, 21:9 (1993), 3197
A. P. Monastyrnyi, V. I. Yanchevskii, “Whitehead groups of spinor groups”, Math. USSR-Izv., 36:1 (1991), 61–100
Bill Jacob, Adrian Wadsworth, “Division algebras over Henselian fields”, Journal of Algebra, 128:1 (1990), 126
Pat Morandi, “The Henselization of a valued division algebra”, Journal of Algebra, 122:1 (1989), 232
Citation in format AMSBIB
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