Kummer subfields of Malcev-Neumann division algebras
type: Item
instance of: scholarly article
title: Kummer subfields of Malcev-Neumann division algebras
zbMATH Open document ID: 0564.16015
DOI: 10.1007/BF02761120
author: Jean-Pierre Tignol
author: S. A. Amitsur
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(Z^ n\) be a free abelian group with a total ordering. The Malcev- Neumann construction with a suitable co-cycle f can then be used to obtain a skew field \(D_ f\) that is finite-dimensional over its centre \(C_ f\). The authors study the Kummer subfields of \(D_ f\) and use this construction to obtain a division algebra, all of whose maximal subfields are Galois over the centre, with elementary abelian Galois group. These facts are applied to obtain the following improvement on the result of \textit{S. A. Amitsur} [ibid. 12, 408-420 (1972; Zbl 0248.16006)]. Let K be an infinite field of characteristic not dividing n and denote by UD(k,n) the universal division algebra of degree n. Write \(n=m'm''\) where \((m',m'')=1\), m'' is square-free, q is the product of the distinct primes dividing m' and \(q^ 2/n\). If H is a splitting group of UD(k,n), then \(| H| \geq n.m'q^{-2}\). This shows e.g. if n is divisible by the cube of a prime, then UD(k,n) is not a crossed product. Another more precise (but more complicated) lower bound is given for the orders of abelian splitting groups of UD(k,n).
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 12E15
zbMATH DE Number: 3899096
zbMATH Keywords: Malcev-Neumann construction
zbMATH Keywords: skew field
zbMATH Keywords: Kummer subfields
zbMATH Keywords: maximal subfields
zbMATH Keywords: elementary abelian Galois group
zbMATH Keywords: universal division algebra
zbMATH Keywords: crossed product
zbMATH Keywords: abelian splitting groups
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
cites work: A note on normal division algebras of prime degree
cites work: On central division algebras
cites work: Division algebras. A survey
cites work: Q3253828
cites work: PI-algebras. An introduction
cites work: Q3882586
cites work: The Second Homology Group of a Group; Relations among Commutators
cites work: On Ordered Division Rings
cites work: Sur l'analysis situs des variétés à \(n\) dimensions
cites work: Cyclic algebras, complete fields, and crossed products
cites work: Dihedral Algebras are Cyclic
cites work: Splittings of Cyclic p-Algebras
cites work: Noncrossed product p-algebras and Galois p-extensions
cites work: Q5796078
cites work: Produits croises Abeliens
cites work: Totally ramified splitting fields of central simple algebras over Henselian fields
cites work: Symplectic modules
cites work: Quadratic forms on finite groups, and related topics
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description: scientific article; zbMATH DE number 3899096