Paare alternierender Formen

Scharlau, Rudolf

doi:10.1007/bf01214270

Cited by 36 publications

(53 citation statements)

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“…(i) bilinear or sesquilinear forms (see, for example, [7,17,18,22]), (ii) pairs of symmetric, or skew symmetric, or Hermitian forms ( [21,23,27,28,29,30]), and (iii) isometric or selfadjoint operators on a space with nondegenerate symmetric, or skew symmetric, or Hermitian form ( [9,10,12,13,22,23]). …”

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confidence: 99%

Classification Problems for Systems of Forms and Linear Mappings

Sergeichuk¹

1988

Math. USSR Izv.

210

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A method is proposed that allow the reduction of many classification problems of linear algebra to the problem of classifying Hermitian forms. Over the complex, real, and rational numbers classifications are obtained for bilinear forms, pairs of quadratic forms, isometric operators, and selfadjoint operators.Many problems of linear algebra can be formulated as problems of classifying the representations of a quiver. A quiver is, by definition, a directed graph. A representation of the quiver is given (see [6], and also [2,14]) by assigning to each vertex a vector space and to each arrow a linear mapping of the corresponding vector spaces. For example, the quivers 1 e e 1 * * 4 4 2 1 9 9 e e correspond respectively to the problems of classifying:• linear operators (whose solution is the Jordan or Frobenius normal form),• pairs of linear mappings from one space to another (the matrix pencil problem, solved by Kronecker), and

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mentioning

confidence: 99%

Classification Problems for Systems of Forms and Linear Mappings

Sergeichuk¹

1988

Math. USSR Izv.

210

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“…If γ : U → W is any linear map then we may create an alternating form on U ⊕ W * by setting and, in this way, a Kronecker module gives rise to a pair of alternating forms on the same space. It is proved in [28] that every pair of alternating forms is equivalent to a pair constructed in this way. The dual of the Kronecker module X = (U, W, α, β) is the Kronecker module X * = (W * , U * , α * , β * ).…”

Section: §1 Introductionmentioning

confidence: 99%

“…In order to prepare for the proof, we must first discuss the classification of pairs of quinary alternating forms over an arbitrary field K. A definitive and beautiful treatment of pairs of alternating forms over an arbitrary field was given by R. Scharlau in [28]. In fact, he reduced the classification problem for such pairs to the classification problem for pairs of rectangular matrices under simultaneous row and column operations.…”

Section: §1 Introductionmentioning

confidence: 99%

“…There is also an obvious notion of the direct sum of Kronecker modules. It is shown in [28] that two Kronecker modules X and Y give rise to equivalent pairs of alternating forms if and only if X ⊕ X * ∼ = Y ⊕ Y * .…”

Section: §1 Introductionmentioning

confidence: 99%

“…In the terminology of [28], a Kronecker module is a quadruple (U, W, α, β) consisting of two K-vector spaces U and W and two K-linear maps α, β : U → W . If γ : U → W is any linear map then we may create an alternating form on U ⊕ W * by setting and, in this way, a Kronecker module gives rise to a pair of alternating forms on the same space.…”

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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