- Title Pages
- Dedication
- Epigraph
- Preface
- Chapter One Buildings
- Chapter Two Quadratic Forms
- Chapter Three Moufang Polygons
- Chapter Four Moufang Quadrangles
- Chapter Five Linked Tori, I
- Chapter Six Linked Tori, II
- Chapter Seven Quadratic Forms over a <span xml:lang="ell">∈</span> Local Field
- Chapter Eight Quadratic Forms of Type E<sub>6</sub>, E<sub>7</sub> and E<sub>8</sub>
- Chapter Nine Quadratic Forms of Type F<sub>4</sub>
- Chapter Ten Residues
- Chapter Eleven Unramified Quadrangles of Type E<sub>6</sub>, E<sub>7</sub> and E<sub>8</sub>
- Chapter Twelve Semi-ramified Quadrangles of Type E<sub>6</sub>, E<sub>7</sub> and E<sub>8</sub>
- Chapter Thirteen Ramified Quadrangles of Type E6, E7 and E8
- Chapter Fourteen Quadrangles of Type E6, E7 and E8: Summary
- Chapter Fifteen Totally Wild Quadratic Forms of Type E7
- Chapter Sixteen Existence
- Chapter Seventeen Quadrangles of Type F4
- Chapter Eighteen The Other Bruhat-Tits Buildings
- Chapter Nineteen Coxeter Groups
- Chapter Twenty Tits Indices
- Chapter Twenty One Parallel Residues
- Chapter Twenty Two Fixed Point Buildings
- Chapter Twenty Three Subbuildings
- Chapter Twenty Four Moufang Structures
- Chapter Twenty Five Fixed Apartments
- Chapter Twenty Six The Standard Metric
- Chapter Twenty Seven Affine Fixed Point Buildings
- Chapter Twenty Eight Pseudo-Split Buildings
- Chapter Twenty Nine Linear Automorphisms
- Chapter Thirty Strictly Semi-linear Automorphisms
- Chapter Thirty One Galois Involutions
- Chapter Thirty Two Unramified Galois Involutions
- Chapter Thirty Three Residually Pseudo-Split Buildings
- Chapter Thirty Four Forms of Residually Pseudo-Split Buildings
- Chapter Thirty Five Orthogonal Buildings
- Chapter Thirty Six Indices for the Exceptional Bruhat-Tits Buildings
- Bibliography
- Index

# Quadratic Forms

# Quadratic Forms

- Chapter:
- (p.13) Chapter Two Quadratic Forms
- Source:
- Descent in Buildings (AM-190)
- Author(s):
### Bernhard M¨uhlherr

### Holger P. Petersson

### Richard M. Weiss

- Publisher:
- Princeton University Press

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.

*Keywords:*
quadratic form, polar space, quadratic module, quadratic space, hyperbolic quadratic module, hyperbolic quadratic space, hyperbolic plane, split quadratic space, round quadratic space

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- Title Pages
- Dedication
- Epigraph
- Preface
- Chapter One Buildings
- Chapter Two Quadratic Forms
- Chapter Three Moufang Polygons
- Chapter Four Moufang Quadrangles
- Chapter Five Linked Tori, I
- Chapter Six Linked Tori, II
- Chapter Seven Quadratic Forms over a <span xml:lang="ell">∈</span> Local Field
- Chapter Eight Quadratic Forms of Type E<sub>6</sub>, E<sub>7</sub> and E<sub>8</sub>
- Chapter Nine Quadratic Forms of Type F<sub>4</sub>
- Chapter Ten Residues
- Chapter Eleven Unramified Quadrangles of Type E<sub>6</sub>, E<sub>7</sub> and E<sub>8</sub>
- Chapter Twelve Semi-ramified Quadrangles of Type E<sub>6</sub>, E<sub>7</sub> and E<sub>8</sub>
- Chapter Thirteen Ramified Quadrangles of Type E6, E7 and E8
- Chapter Fourteen Quadrangles of Type E6, E7 and E8: Summary
- Chapter Fifteen Totally Wild Quadratic Forms of Type E7
- Chapter Sixteen Existence
- Chapter Seventeen Quadrangles of Type F4
- Chapter Eighteen The Other Bruhat-Tits Buildings
- Chapter Nineteen Coxeter Groups
- Chapter Twenty Tits Indices
- Chapter Twenty One Parallel Residues
- Chapter Twenty Two Fixed Point Buildings
- Chapter Twenty Three Subbuildings
- Chapter Twenty Four Moufang Structures
- Chapter Twenty Five Fixed Apartments
- Chapter Twenty Six The Standard Metric
- Chapter Twenty Seven Affine Fixed Point Buildings
- Chapter Twenty Eight Pseudo-Split Buildings
- Chapter Twenty Nine Linear Automorphisms
- Chapter Thirty Strictly Semi-linear Automorphisms
- Chapter Thirty One Galois Involutions
- Chapter Thirty Two Unramified Galois Involutions
- Chapter Thirty Three Residually Pseudo-Split Buildings
- Chapter Thirty Four Forms of Residually Pseudo-Split Buildings
- Chapter Thirty Five Orthogonal Buildings
- Chapter Thirty Six Indices for the Exceptional Bruhat-Tits Buildings
- Bibliography
- Index