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Noether’s problem and normalization. (English) Zbl 1055.12501
The author provides a very well-written survey of the current status of the work on Noether’s problem. Let \(k\) be a field, \(k(x_1,\cdots,x_n)\) the rational function field in \(n\) variables, and \(G\) a finite group acting as automorphisms of \(k(x_1,\cdots,x_n)\) and permuting the variables, then the question studied is whether the fixed field of the automorphisms is also a rational function field in \(n\) variables. She nicely presents the work of several researchers, a. o. Noether, Saltman, Swan, Lenstra, Colliot-Thélène, Sansuc, up to the present state. A large bibliography of 85 items is added.

12F10 Separable extensions, Galois theory
12F20 Transcendental field extensions
01A60 History of mathematics in the 20th century