Noether’s problem for dihedral 2-groups II

Abstract

is rational (= purely transcendental) over K . A result of Serre shows that ‫(ޑ‬G) is not rational when G is the generalized quaternion group of order 16. We shall prove that K (G) is rational over K if G is any nonabelian group of order 16 except when G is the generalized quaternion group of order 16. When G is the generalized quaternion group of order 16 and K (ζ 8 ) is a cyclic extension of K , then K (G) is also rational over K .

Noether's problem for four- and five-dimensional linear actions

“…The following theorem is a result for any field, but if we apply it for subgroups of GL(4, Q), we get the partial results of Problem 3 for K = Q and n = 4. [11].) Problem 3 for K = Q and n = 4 has an affirmative answer for any non-abelian groups of order 16, except for one which is isomorphic to the generalized quaternion group.…”

“…1 If G is the generalized quaternion group of order 16 and K (ζ 8 ) is cyclic over K (in case char K = 2), then Noether's problem is also affirmative. See Theorem 1.4 in [11].…”

Noether's problem for groups of order 32

“…(See [Ka1].) Let G be a metacyclic p-group of exponent e, and K be a field containing a primitive eth root of unity.…”

Noether’s problem forŜ4andŜ5

“…In case G is the generalized quaternion group of order 16 and k. 8 / is cyclic over k, it is known that k.G/ is k-rational [Kang 2005]. We don't know whether analogous results as Theorem 1.4 are valid when the groups are SL 2 ‫ކ.‬ 7 / and SL 2 ‫ކ.‬ 9 /.…”