Prehomogeneous vector spaces and field extensions

Wright, D. J.; Yukie, Akihiko

doi:10.1007/bf01231334

Cited by 80 publications

(135 citation statements)

References 15 publications

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“…One of the cases Bhargava considered was the space of pairs of ternary quadratic forms. This case is one of the cases considered in [32] and orbits over fields correspond to quartic extensions of the ground field. In [2], Bhargava constructed a bijective map from the set of integral orbits of this prehomogeneous vector space to the set of triples (R, C, f ) where R is a quartic ring, C is its resolvent ring (this notion of the resolvent ring is due to him) and f is a certain quadratic map from R to C. If one only considers R, this gives a map from the set of integral orbits of pairs of ternary quadratic forms to the set of quartic rings, thus giving an analogue of Delone and Fadeev's map.…”

Section: §1 Introductionmentioning

confidence: 99%

“…Among the eight examples considered in [32], two were classical and six were new, at least in this form. This list included the space of pairs of ternary quadratic forms, parameterizing quartic algebras over the ground field and the space of quadruples of quinary alternating forms, parameterizing quintic algebras over the ground field even though the ring structures were not given directly.…”

Section: §1 Introductionmentioning

confidence: 99%

“…In [32], orbits of eight prehomogeneous vector spaces over fields were considered and the interpretations of orbits of these prehomogeneous vector spaces were given. As far as integral orbits are concerned, besides classical works, [15] was the first work that was carried out in the context of prehomogeneous vector spaces.…”

Section: §1 Introductionmentioning

confidence: 99%

“…As far as integral orbits are concerned, besides classical works, [15] was the first work that was carried out in the context of prehomogeneous vector spaces. In [15], integral orbits in one of the smaller examples identified in [32] were considered. Bhargava [2] considered integral orbits of several important cases.…”

Section: §1 Introductionmentioning

confidence: 99%

“…This list included the space of pairs of ternary quadratic forms, parameterizing quartic algebras over the ground field and the space of quadruples of quinary alternating forms, parameterizing quintic algebras over the ground field even though the ring structures were not given directly. The correspondences in [32] were carried out in two ways, via geometry and via Galois cohomology. It is also natural to attempt to use invariant-theoretic methods to construct orbit-ring maps and this method was used in Section 2 of [31] to construct an orbit-ring map over a field with parameter space the space of trivectors on sevendimensional affine space, parameterizing Cayley algebras over the field.…”

Section: §1 Introductionmentioning

confidence: 99%

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A construction of quintic rings

Kable¹

2004

Nagoya Mathematical Journal

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Abstract. We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant. §1. IntroductionTo set the stage, we shall first remind the reader of the work of Delone and Fadeev. In [9], these authors began with an integral binary cubic form f and showed how to construct from it a based cubic ring R f over Z. (Here, as below, an n-tic ring over Z is a commutative Z-algebra that is free of rank n as a Z-module. A ring is based if it is equipped with a distinguished basis containing 1.) They showed that the isomorphism class of R f as a ring depends only upon the GL(2, Z)-equivalence class of f and that the map [f ] → [R f ] from the set of equivalence classes of non-singular forms to the set of isomorphism classes of separable cubic rings is bijective. Moreover, they showed that their map is discriminant-preserving, where the discriminant of f is understood in the sense of invariant theory and that of R f in the sense of ring theory. It is also (probably more) well known that Gauss constructed a correspondence between the set of equivalence

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Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%