zbMATH — the first resource for mathematics

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 \(p_ x= rp_ y r^{-1}\).

12F10 Separable extensions, Galois theory
11S90 Prehomogeneous vector spaces
14M17 Homogeneous spaces and generalizations
Full Text: DOI EuDML
[1] Brioschi, F.: Sulla risoluzione della equazioni del quinto grado. In: Opere Mathematiche, Vol. I, pp. 335-341. Milano: Ulrico Hoepi 1904
[2] Datskovsky, B.: A mean value theorem for class numbers of quadratic extensions. In: The memorial volume for Emil Grosswald, 1992 (to appear)
[3] Datskovsky, B., Wright, D.J.: The adelic zeta function associated with the space of binary cubic forms, II: Local theory. J. Reine Angew. Math.367, 27-75 (1986) · Zbl 0575.10016 · doi:10.1515/crll.1986.367.27
[4] Datskovsky, B., Wright, D.J.: Density of discriminants of cubic extensions. J. Reine Angew. Math.386, 116-138 (1988) · Zbl 0632.12007 · doi:10.1515/crll.1988.386.116
[5] Davenport, H.: On the class-number of binary cubic forms. I and II J. Lond. Math. Soc.26, 183-198, 1951. Corrigendum: ibid Davenport, H.: On the class-number of binary cubic forms. I and II J. Lond. Math. Soc. 27, 512 (1952) · Zbl 0044.27002 · doi:10.1112/jlms/s1-26.3.183
[6] Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields I. Bull. Lond. Math. Soc.1, 345-348 (1961) · Zbl 0211.38602 · doi:10.1112/blms/1.3.345
[7] Davenport, H. and Heilbronn, H.: On the density of discriminants of cubic fields. II. Proc. R. Soc. Lond., Sci.A322, 405-420 (1971) · Zbl 0212.08101 · doi:10.1098/rspa.1971.0075
[8] Goldfeld, D., Hoffstein, J.: Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet series. Invet. Math.80, 185-208 (1985) · Zbl 0564.10043 · doi:10.1007/BF01388603
[9] Hermite, C.: Sur la résolution de l’équation du cinquième degré. In: Oeuvres, Vol. II, pp. 5-12. Paris; Gauthier-Villars 1908
[10] Igusa, J.: On a certain class of prehomogeneous vector spaces. J. Pure Appl. Algebra47, 265-282 (1987) · Zbl 0633.14030 · doi:10.1016/0022-4049(87)90051-X
[11] Klein, F.: Weitere Untersuchungen über das Ikosaeder. In: Abhandlungen, Vol. II, pp. 321-384, Berlin: Springer 1992
[12] Klein, F.: Lectures on the icosahedron and the solution of equations of the fifth degree. New York: Dover 1956 · Zbl 0072.25901
[13] Kneser, M.: Lectures on Galois cohomology of classical groups. Tata Lecture Notes. Bombay 1969 · Zbl 0246.14008
[14] Kronecker, L.: Sur la résolution de l’équation du cinquième degré (extract d’une letter addressée à M. Hermite). In: Werke, Hensel, K. (ed.) Vol. IV, pp. 43-47. Leipzig Berlin: Teubner 1929
[15] Rubenthaler, H.: Espaces préhomogenes de type parabolique. Thèse. Université Strasbourg, 1982 · Zbl 0546.22019
[16] Rubenthaler, H., Schiffmann, G.: Opérateurs differéntiels de Shimura et espaces préhomogènes. Invent. Math.90, 409-442 (1987) · Zbl 0638.10024 · doi:10.1007/BF01388712
[17] Sato, M., Kimura, T.: A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J.65, 1-155 (1977) · Zbl 0321.14030
[18] Sato, M., Shintani, T.: On zeta functions associated with prehomogeneous vector spaces. Ann. Math.100, 131-170 (1974) · Zbl 0309.10014 · doi:10.2307/1970844
[19] Serre, J.P.: Cohomologie Galoisienne (Lect. Notes Math., Vol. 5) Berlin Heidelberg New York: Springer 1965
[20] Serre, J.P.: Extensions icosaédrique. In: Oeuvres, Vol. III, p. 550-554. Springer, Berlin, Heidelberg, New York 1986
[21] Shintani, T.: On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms. J. Math. Soc. Japan,24, 132-188 (1972) · Zbl 0227.10031 · doi:10.2969/jmsj/02410132
[22] Shintani, T.: On zeta-functions associated with vector spaces of quadratic forms. J. Fac. Sci. Univ. Tokyo, Sect IA,22, 25-66 (1975) · Zbl 0313.10041
[23] Suzuki, M.: Group theory, Vol. I. Springer, Berlin, Heidelberg, New York, 1982. · Zbl 0472.20001
[24] Van der Waerden, B.: History of algebra. Springer, Berlin, Heidelberg, New York. 1985 · Zbl 0569.01001
[25] Vinberg, E.B.: On the classification of the nilpotent elements of graded lie algebras. Sov. Math. Dokl.,16, 1517-1520 (1975) · Zbl 0374.17001
[26] Wright, D.J.: The adelic zeta function associated to the space of binary cubic forms. part i: Global theory. Math. Ann.270, 503-534 (1985) · Zbl 0545.10014 · doi:10.1007/BF01455301
[27] Yukie, A.: On the Shintani zeta function for the space of binary quadratic forms. Math. Ann.292, 355-374 (1992) · Zbl 0757.11027 · doi:10.1007/BF01444626
[28] Yukie, A.: Shintani zeta functions (Preprint 1992) · Zbl 0757.11027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.