Rationality problem of GL4 group actions
Journal
Advances in Mathematics
Journal Volume
181
Journal Issue
2
Pages
321-352
Date Issued
2004
Author(s)
Abstract
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. Both K(V)〈σ〉 and K(ℙ(V))〈σ〉 are rational (= purely transcendental) over K if at least one of the following conditions is satisfied: (i) char K = 2, (ii) f(T) is a reducible or inseparable polynomial in K[T], (iii) not all of β1,β2,β3 are roots of unity, (iv) if f(T) is separable irreducible, then the Galois group of f(T) over K is not isomorphic to the dihedral group of order 8 or the Klein four group. Theorem 2. Suppose that all βi are roots of unity and f(T) ∈ K[T] is separable irreducible. (a) If the Galois group of f(T) is isomorphic to the dihedral group of order 8, then both K(V)〈σ〉 and K(ℙ(V) 〈σ〉 are not stably rational over K. (b) When the Galois group of f(T) is isomorphic to the Klein four group, then a necessary and sufficient condition for rationality of K(V)〈σ〉 and K(ℙ(V)〈σ〉 is provided. (See Theorem 1.5. for details.) © 2003 Elsevier Science (USA). All rights reserved.
Type
journal article