## Quadratic Forms with Applications to Algebraic Geometry and TopologyA gem of a book bringing together 30 years worth of results that are certain to interest anyone whose research touches on quadratic forms. |

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### Contents

Chapter 1 The Representation Theorems of Cassels | 1 |

Chapter 2 Multiplicative Quadratic Forms | 19 |

Chapter 3 The Level of Fields Rings and Topological Spaces | 40 |

Chapter 4 Hilberts Homogeneous Nullstellensatz forpfields and Applications to Topology | 51 |

Chapter 5 TsenLang Theory for Cffields | 64 |

Chapter 6 Hilberts 17th Problem | 75 |

Chapter 7 The Pythagoras Number | 94 |

Chapter 8 The itinvariant | 110 |

Chapter 9 Systems of Quadratic Forms | 132 |

Chapter 10 The Level of Projective Spaces | 153 |

166 | |

List of Symbols | 176 |

178 | |

### Common terms and phrases

addition algebraic algebraic topology anisotropic apply arbitrary assume called Chapter char clear Clearly closed closure Compare complete condition Conjecture consider contains contradiction Corollary deﬁned Deﬁnition denote dimension element equation equivalence estimate Example exists extension ﬁeld K ﬁnite ﬁrst form go given gives go represents hence holds ideal immediately implies induced integral interesting invariant involution isotropic Lemma linear Math matrix maximal natural nonreal nontrivial Note Open orthogonal particular polynomial positive possible prime problem proof properties Proposition proved Pythagoras number quadratic form quadratic map question real ﬁeld regular relation replaced respect result ring satisﬁes shows similar space sum of squares Suppose symmetric matrix Theorem theory topology trivial u-invariant unique universal variables vector Witt zero