TY - JOUR

T1 - Noether's problem and unramified brauer groups

AU - Hoshi, Akinari

AU - Kang, Ming Chang

AU - Kunyavskii, Boris E.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Bogomolov multipliers

KW - Noether's problem

KW - Rationality

KW - Rationality problem

KW - Retract rationality

KW - Unramified Brauer groups

UR - http://www.scopus.com/inward/record.url?scp=84884952911&partnerID=8YFLogxK

U2 - 10.4310/AJM.2013.v17.n4.a8

DO - 10.4310/AJM.2013.v17.n4.a8

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AN - SCOPUS:84884952911

SN - 1093-6106

VL - 17

SP - 689

EP - 714

JO - Asian Journal of Mathematics

JF - Asian Journal of Mathematics

IS - 4

ER -