Math 2.0

Associative rings and algebras Division rings and central simple algebras Counterexample

Amitsur's noncrossed product

Builds on Division ringCentral simple algebraDegree of a central simple algebraCrossed product algebraGeneric matrix algebra

Counterexample

Let k=ℚk = \Q and let n≥2n \ge 2. Let XX and YY be two n×nn \times n matrices whose entries are independent commuting variables over kk. The kk-algebra generated by XX and YY is the generic matrix algebra; it has no zero divisors. Its ring of central quotients UD(k,n)\mathrm{UD}(k, n) is the generic division algebra. It is a division ring and a central simple algebra of degree nn over its center. Amitsur proved that UD(k,n)\mathrm{UD}(k, n) is not a crossed product algebra when nn is divisible by 88 or by p2p^2 for an odd prime pp. That is, for such nn it contains no maximal subfield that is a Galois extension of its center.

Remarks

The idea of the proof is to specialize. A crossed product structure on UD(k,n)\mathrm{UD}(k, n) with group GG would pass to every division algebra of degree nn over suitable fields of iterated Laurent series. Amitsur exhibited two such division algebras that are crossed products only for groups of incompatible types, so no single group GG works. For small degrees the opposite holds: every central division algebra of degree 22, 33, 44 or 66 is a crossed product. Over local and global fields every central simple algebra is cyclic, so noncrossed products can only appear over more complicated fields.

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To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Central simple algebra, concept, no evidence levelCentral simple algebraconceptCrossed product algebra, concept, no evidence levelCrossed product algeb…conceptDegree of a central simple algebra, concept, no evidence levelDegree of a central s…conceptDivision ring, concept, no evidence levelDivision ringconceptGeneric matrix algebra, concept, no evidence levelGeneric matrix algebraconceptAmitsur's noncrossed product, counterexample, no evidence levelAmitsur's noncrossed …counterexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Central simple algebra concept
  2. Crossed product algebra concept
  3. Degree of a central simple algebra concept
  4. Division ring concept
  5. Generic matrix algebra concept
  6. Amitsur's noncrossed product counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Crossed_product
About this page
Tag8A00Y84SW7
URIm2://8A00Y84SW7
Kindcounterexample
WritingW0 stub
MSC16K20, 16S35, 16R20
UsesDivision ring, Central simple algebra, Degree of a central simple algebra, Crossed product algebra, Generic matrix algebra
Changed2 November 2026

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