Noether's problem and unramified brauer groups

Akinari Hoshi, Ming Chang Kang, Boris E. Kunyavskii

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Let k be any field, G be a finite group acting on the rational function field k(xg : g ε G) by h · xg = xhg for any h, g ε G. Define k(G) = k(xg : g ε G)G. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is known that, if ℂ(G) is rational over ℂ, then B0(G) = 0 where B0(G) is the unramified Brauer group of ℂ(G) over ℂ. Bogomolov showed that, if G is a p-group of order p5, then B0(G) = 0. This result was disproved by Moravec for p = 3, 5, 7 by computer calculations. We will prove the following theorem. Theorem. Let p be any odd prime number, G be a group of order p5. Then B0(G) ≠ 0 if and only if G belongs to the isoclinism family π10 in R. James's classification of groups of order p5.

Original languageEnglish
Pages (from-to)689-714
Number of pages26
JournalAsian Journal of Mathematics
Issue number4
StatePublished - 2013


  • Bogomolov multipliers
  • Noether's problem
  • Rationality
  • Rationality problem
  • Retract rationality
  • Unramified Brauer groups


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