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**Trilinear alternating forms on a vector space of dimension 7.**
*(English)*
Zbl 0646.15019

If E is a vector space of dimension n over the field F, then a trilinear form f on E is called alternating, if for all \(x_ 1,x_ 2,x_ 3\in E\) with \(x_ i=x_ j\) for at least two distinct i,j (1\(\leq i,j\leq 3)\) we have \(f(x_ 1,x_ 2,x_ 3)=0\). The classification of these forms was carried out by J. A. Schouten [Rend. Circ. Mat. Palermo 55, 137-156 (1931; Zbl 0001.35401)] for \(F={\mathbb{C}}\) (complex numbers) and by J. Cresp in his Thesis (1976) for F algebraically closed of characteristic \(\neq 2,3\) and \(n=7.\)

In the underlying paper a short proof is given of the classification for F algebraically closed of arbitrary characteristic and \(n=7\). Moreover a classification for \(n\leq 7\) follows over certain non-algebraically closed fields, including all finite fields by use of Galois cohomology.

In the underlying paper a short proof is given of the classification for F algebraically closed of arbitrary characteristic and \(n=7\). Moreover a classification for \(n\leq 7\) follows over certain non-algebraically closed fields, including all finite fields by use of Galois cohomology.

Reviewer: A.H.Boers

### MSC:

15A63 | Quadratic and bilinear forms, inner products |

11E76 | Forms of degree higher than two |

20G15 | Linear algebraic groups over arbitrary fields |

17D05 | Alternative rings |

### Keywords:

alternating forms; dimension 7; trilinear forms on a vector space; classification; non-algebraically closed fields; finite fields; Galois cohomology### Citations:

Zbl 0001.35401### References:

[1] | Borel A., Linear Algebraic Groups (1969) · Zbl 0186.33201 |

[2] | Bourbaki N., Elements de Mathématique Algèbre (1971) |

[3] | Cresp J., Orbits in , Kummer manifolds and the cohomology of a hyperplane section of a Grassmannian, Thesis (1976) |

[4] | Dieudonné J., La géométrie des groupes classiques (1955) · Zbl 0067.26104 |

[5] | Gurevich G.B., Foundations of the Theory of Algebraic Invariants (1964) · Zbl 0128.24601 |

[6] | Lang S., Algebra (1966) |

[7] | DOI: 10.1007/BF03016791 · Zbl 0001.35401 · doi:10.1007/BF03016791 |

[8] | Serre J.P., Lecture Notes in Math 5 (1964) |

[9] | Springer T.A., Oktaven, Jordan-Algebren und Ausnahmegruppen (1963) |

[10] | Migliore, Phd. Thesis (1982) |

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