On nonsingular bilinear maps over different fields

Research Article
Open access

On nonsingular bilinear maps over different fields

Ruoxuan Feng 1*
  • 1 WLSA Shanghai Academy    
  • *corresponding author reinafeng2006@163.com
Published on 17 November 2023 | https://doi.org/10.54254/2753-8818/11/20230417
TNS Vol.11
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-133-9
ISBN (Online): 978-1-83558-134-6

Abstract

The investigation of nonsingular bilinear forms originates from the classification of division algebras over the real number field. Building upon this foundation, researchers have delved into the study of nonsingular bilinear forms over real number fields, leading to significant results such as Hopf’s theorem. However, the interest in understanding nonsingular bilinear forms extends beyond real number fields, prompting a desire to explore other fields as well. When it comes to algebraically closed fields, the theorem becomes well-understood, with essence captured by the Hopf-Smith theorem. Inspired by these established studies, we are motivated to further the comprehension of nonsingular bilinear forms over arbitrary fields. Given a field , we study in this article numerical constraints on for the existence of nonsingular bilinear maps for not only algebraically closed fields and the real number field but also the rational number field and finite fields. We reach the final conclusion mainly through algebro-geometric techniques and the use of determinantal varieties. We reprove a result of Hopf–Smith which states that the minimal possible value of is when is an algebraically closed field. When is the real number field, we prove that under a combinatorial condition, the minimal possible value of is still . We also show that when is the rational number field or a finite field, the minimal possible value of is .

Keywords:

nonsingular bilinear forms, Hopf–Smith theorem, determinantal varieties, field extensions.

Feng,R. (2023). On nonsingular bilinear maps over different fields. Theoretical and Natural Science,11,256-271.
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References

[1]. Daniel B. Shapiro, Compositions of Quadratic Forms, 2000, de Gruyter Expositions in Mathematics 33.

[2]. H. Hopf, Ein topo1ogischer Beitrag zur reellen Algebra, Comment. Math. Helv. 13 (1940/41), 219-239.

[3]. L. Smith, Nonsingular bilinear forms, generalized J homomorphisms, and the homotopy of spheres I, 1978, Indiana Univ. Math. J. 27, 697–737.

[4]. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves, Volume I, 1985, Springer.

[5]. D. Mumford, The Red Book of Varieties and Schemes, 1974, Lecture notes in mathematics 1358, Springer.

[6]. J. Harris, L. W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology, Volume 23, Issue 1, 1984, Pages 71-84.

[7]. D. Huybrechts, Complex Geometry, An Introduction, Univertext, Springer.

[8]. M. Reid, Undergraduate Algebraic Geometry, London Mathematical Society Student Texts 12, Cambridge University Press.

[9]. M. Artin, Algebra, Pearson Prentice Hall, 2011.


Cite this article

Feng,R. (2023). On nonsingular bilinear maps over different fields. Theoretical and Natural Science,11,256-271.

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The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation

ISBN:978-1-83558-133-9(Print) / 978-1-83558-134-6(Online)
Editor:Roman Bauer
Conference website: https://www.confmpcs.org/
Conference date: 12 August 2023
Series: Theoretical and Natural Science
Volume number: Vol.11
ISSN:2753-8818(Print) / 2753-8826(Online)

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References

[1]. Daniel B. Shapiro, Compositions of Quadratic Forms, 2000, de Gruyter Expositions in Mathematics 33.

[2]. H. Hopf, Ein topo1ogischer Beitrag zur reellen Algebra, Comment. Math. Helv. 13 (1940/41), 219-239.

[3]. L. Smith, Nonsingular bilinear forms, generalized J homomorphisms, and the homotopy of spheres I, 1978, Indiana Univ. Math. J. 27, 697–737.

[4]. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves, Volume I, 1985, Springer.

[5]. D. Mumford, The Red Book of Varieties and Schemes, 1974, Lecture notes in mathematics 1358, Springer.

[6]. J. Harris, L. W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology, Volume 23, Issue 1, 1984, Pages 71-84.

[7]. D. Huybrechts, Complex Geometry, An Introduction, Univertext, Springer.

[8]. M. Reid, Undergraduate Algebraic Geometry, London Mathematical Society Student Texts 12, Cambridge University Press.

[9]. M. Artin, Algebra, Pearson Prentice Hall, 2011.