Rationality problem of GL4 group actions

Let 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〉andare 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 allβ_iare 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〉andare 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〉andis provided. (See Theorem 1.5. for details.)