Fields of definition for division algebras

M. Lorenz, Z. Reichstein, L. H. Rowen, D. J. Saltman

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21 Scopus citations

Abstract

Let A be a finite-dimensional division algebra containing a base field k in its center F. A is defined over a subfield F0 if there exists an F0-algebra A0 such that A = A0 ⊗F 0 F. The following are shown, (i) In many cases A can be defined over a rational extension of k. (ii) If A has odd degree n ≥ 5, then A is defined over a field F0 of transcendence degree ≤ ≤1/2(n - 1) (n - 2) over k. (iii) If A is a ℤ/m × ℤ/2-crossed product for some m ≥ 2 (and in particular, if A is any algebra of degree 4) then A is Brauer equivalent to a tensor product of two symbol algebras. Consequently, Mm(A) can be defined over a field F0 such that trdeg k(F0) ≤ 4. (iv) If A has degree 4 then the trace form of A can be defined over a field F0 of transcendence degree ≤ 4. (In (i), (iii) and (iv) it is assumed that the center of A contains certain roots of unity.).

Original languageEnglish
Pages (from-to)651-670
Number of pages20
JournalJournal of the London Mathematical Society
Volume68
Issue number3
DOIs
StatePublished - Dec 2003
Externally publishedYes

Bibliographical note

Funding Information:
M. Lorenz was supported in part by NSF grant DMS-9988756. Z. Reichstein was supported in part by NSF grant DMS-9801675 and an NSERC research grant. L. H. Rowen was supported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities – Center of Excellence program 8007/99-3. D. J. Saltman was supported in part by NSF grant DMS-9970213.

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