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Indecomposable division algebras with a Baer ordering. (English) Zbl 0815.16026

The authors’ object is to construct Baer ordered indecomposable division algebras [cf. R. Baer, Linear Algebra and Projective Geometry (Academic Press 1952; Zbl 0049.381)] which are of index \(p^ n\) and exponent \(p^ m\) for all primes \(p\) and all integers \(n\), \(m\) satisfying \(n > m \geq 1\) (where \(n \geq 3\) if \(p = 2\)). In particular this leads to Baer ordered indecomposable division algebras of prime exponent and indecomposable division algebras that are Baer ordered with respect to an involution of the first kind. The first step is to construct an indecomposable division algebra of index \(p^ n\) and exponent \(p\) (or exponent 4 if \(p = 2\)) for any prime \(p\) and any integer \(n \geq 2\), with subfields of certain specified properties. Next the ground field is extended by indeterminates, its Henselization with respect to a certain valuation is formed and in an appropriate tensor product the underlying division algebra provides the desired example. The paper is a sequel to a paper by the authors (to appear) in which they construct Baer ordered noncrossed product division algebras of index \(p^ n\) and exponent \(p^ m\) for all primes \(p\) and all \(n \geq m \geq 3\).
Reviewer: P.M.Cohn (London)

MSC:

16W80 Topological and ordered rings and modules
16K20 Finite-dimensional division rings
12J15 Ordered fields
12J20 General valuation theory for fields
16W10 Rings with involution; Lie, Jordan and other nonassociative structures

Citations:

Zbl 0049.381
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References:

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