On nonsingular bilinear maps over different fields

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 Hopfs 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 HopfSmith 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 .