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Prehomogeneous vector spaces and field extensions. (English) Zbl 0803.12004
The purpose of this paper is to give a correspondence between the set of orbits of certain prehomogeneous vector spaces and the set of isomorphism classes of Galois extensions of a field \(k\). Let \(k\) be an infinite field of characteristic \(\neq 2,3,5\). The set \({\mathfrak {Gr}}_ i\) is, by definition, the set of isomorphism classes of Galois extensions of \(k\) which are splitting fields of degree \(i\) equations without multiple roots. Let \((G_ k, V_ k)\) be a prehomogeneous vector space with a relative invariant \(\Delta(x)\). We set \(V_ k^{ss}:= \{x\in V_ k\); \(\Delta(x)\neq 0\}\). For \(2\leq i\leq 5\), the authors construct a natural map \(\alpha_ V: G_ k \setminus V_ k^{ss}\ni x\mapsto k(x)\in {\mathfrak {Gr}}_ i\) for certain prehomogeneous vector space \((G_ k, V_ k)\). For each point \(x\in V_ k^{ss}\), they associate a homomorphism \(p_ k: \text{Gal} (k^{\text{sep}}/k) \to{\mathfrak S}_ i\) where \(\text{Gal}(-)\) stands for the Galois group of separable extensions over \(k\) and \({\mathfrak S}_ i\) is the permutation group of degree \(i\). The main result of this paper is the following: let \(x,y\in V_ k^{ss}\) and \(k(x)= k(y)= k'\in {\mathfrak {Gr}}_ i\). Then \(x\) and \(y\) are \(G_ k\)- equivalent if and only if there exists \(r\in{\mathfrak S}\) such that \(p_ x= rp_ y r^{-1}\).

MSC:
12F10 Separable extensions, Galois theory
11S90 Prehomogeneous vector spaces
14M17 Homogeneous spaces and generalizations
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