Noether's problem for p-groups of order p⁵

2013/09/02 by Chen, Yin
#Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1309.0413

Abstract

Let k be any field, p>3 be any prime number and G be a nonabelian p-group of order p⁵. Consider the action of G on the rational function field k(x_h:h∈ G) by g⋅ x_h=x_gh for all g,h∈ G. Let e be the exponent of G. Noether's problem asks whether the fixed field k(G)=k(x_h:h∈ G)^G is rational (i.e., purely transcendental) over k. In this paper, we will prove that if G does not belong to the isoclinic family Φ₁₀ in James's classification \citeJam1980 and k contains a primitive eth root of unity, then k(G) is rational over k. As a corollary, if k=C is the field of complex numbers, then C(G) is rational over C if and only if G is not in the family Φ₁₀. This refines a recent result of Hoshi, Kang and Kunyavskii (\citeHKK2012, Theorem 1.12).

Related

• Noether's problem for p-groups with three generators
• Noether's problem for some p-groups
• Noether's problem for p-groups with an abelian subgroup of index p
• Noether's problem for p_groups with a cyclic subgroup of index p²
• Noether's problem and unramified Brauer groups