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- W2011882525 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 , αβ 1 , αβ 2 , αβ 3 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 β 1 , β 2 , β 3 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 β i 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.)" @default.
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- W2011882525 date "2004-01-01" @default.
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- W2011882525 title "Rationality problem of GL4 group actions" @default.
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- W2011882525 doi "https://doi.org/10.1016/s0001-8708(03)00067-7" @default.
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