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A note on quadratic forms

Fabian Hebestreit, Achim Krause, Maxime Ramzi

Abstract

For a field extension $L/K$ we consider maps that are quadratic over $L$ but whose polarisation is only bilinear over $K$. Our main result is that all such are automatically quadratic forms over $L$ in the usual sense if and only if $L/K$ is formally unramified. In particular, this shows that over finite and number fields, one of the axioms in the standard definition of quadratic forms is superfluous.

A note on quadratic forms

Abstract

For a field extension we consider maps that are quadratic over but whose polarisation is only bilinear over . Our main result is that all such are automatically quadratic forms over in the usual sense if and only if is formally unramified. In particular, this shows that over finite and number fields, one of the axioms in the standard definition of quadratic forms is superfluous.
Paper Structure (4 sections, 13 theorems, 91 equations)

This paper contains 4 sections, 13 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main results
  4. Three examples

Key Result

Corollary 1

The third axiom in the definition of an $L$-quadratic form is superfluous whenever $L$ is an algebraic field extension of a prime field, e.g. when $L$ is a finite or a number field.

Theorems & Definitions (27)

  • Definition
  • Corollary
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • ...and 17 more