Let be any field and be a finite group. Let act on the rational function field by -automorphisms defined by for any . Denote by the fixed field . Noethers problem asks whether is rational (= purely transcendental) over . We shall prove that is rational over if is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.
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Huah Chu, Ming-Chang Kang, Shou-Jen Hu, Noether's problem for dihedral 2-groups. Comment. Math. Helv. 79 (2004), no. 1, pp. 147–159DOI 10.1007/S00014-003-0783-8