Unramified cohomology of degree 3 and Noether’s problem

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W) ^G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W) ^G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G. In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W) ^G . Specializing to the case where G is a central extension of an F _p -vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W) ^G is not rational although its unramified cohomology group of degree 2 is trivial. Dedicated to Jean-Louis Colliot-Thélène.