Rationality problem of GL4 group actions

AbstractLet K be any field which may not be algebraically closed, V be a four-dimensional vector space over K, σ∈GL(V) where the order of σ may be finite or infinite, f(T)∈K[T] be the characteristic polynomial of σ. Let α, αβ1, αβ2, αβ3 be the four roots of f(T)=0 in some extension field of K.Theorem 1.BothK(V)〈σ〉andK(P(V))〈σ〉are rational (=purelytranscendental) overKif at least one of the following conditions is satisfied: (i) charK=2, (ii) f(T) is a reducible or inseparable polynomial inK[T], (iii) not all ofβ1,β2,β3are roots of unity, (iv) iff(T) is separable irreducible, then the Galois group off(T) overKis not isomorphic to the dihedral group of order 8 or the Klein four group.Theorem 2.Suppose that allβiare roots of unity andf(T)∈K[T] is separable irreducible. (a) If the Galois group off(T) is isomorphic to the dihedral group of order 8, then bothK(V)〈σ〉andK(P(V))〈σ〉are not stably rational overK. (b) When the Galois group off(T) is isomorphic to the Klein four group, then a necessary and sufficient condition for rationality ofK(V)〈σ〉andK(P(V))〈σ〉is provided. (See Theorem 1.5. for details.