Canonical matrices of forms and pairs of forms over finite and p-adic fields

TL;DR

This paper provides canonical matrix forms for bilinear, sesquilinear, and pairs of symmetric, skew-symmetric, or Hermitian forms over finite and p-adic fields, extending previous classifications to these specific fields.

Contribution

It extends the classification of canonical matrices for forms and pairs of forms to finite fields of characteristic not 2 and p-adic fields, covering new cases not previously addressed.

Findings

Canonical matrices for forms over finite fields are established.

Canonical matrices for forms over p-adic fields are characterized.

The results extend prior classifications to specific non-algebraically closed fields.

Abstract

Canonical matrices of (a) bilinear and sesquilinear forms, (b) pairs of forms, in which every form is symmetric or skew-symmetric, and (c) pairs of Hermitian forms are given over finite fields of characteristic not 2 and over finite extensions of the field Q_p of p-adic numbers with p not 2. These canonical matrices are special cases of the canonical matrices of (a)-(c) over a field of characteristic not 2 that were obtained in [V. V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501] up to classification of quadratic or Hermitian forms over its finite extensions.