## Introduction to Quadratic FormsFrom the reviews: "O'Meara treats his subject from this point of view (of the interaction with algebraic groups). He does not attempt an encyclopedic coverage ...nor does he strive to take the reader to the frontiers of knowledge... . Instead he has given a clear account from first principles and his book is a useful introduction to the modern viewpoint and literature. In fact it presupposes only undergraduate algebra (up to Galois theory inclusive)... The book is lucidly written and can be warmly recommended. J.W.S. Cassels, The Mathematical Gazette, 1965 "Anyone who has heard O'Meara lecture will recognize in every page of this book the crispness and lucidity of the author's style;... The organization and selection of material is superb... deserves high praise as an excellent example of that too-rare type of mathematical exposition combining conciseness with clarity... R. Jacobowitz, Bulletin of the AMS, 1965 |

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### Contents

II | 1 |

III | 14 |

IV | 20 |

V | 28 |

VI | 30 |

VII | 37 |

VIII | 41 |

IX | 42 |

XXVIII | 157 |

XXIX | 158 |

XXX | 172 |

XXXI | 173 |

XXXII | 186 |

XXXIII | 190 |

XXXV | 203 |

XXXVI | 205 |

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### Common terms and phrases

algebra anisotropic apply archimedean assume base claim clearly commutative complete condition Consider consists contains Corollary Dedekind define definition denote determined discrete easily element equal equation equivalent Example expressed extension fact factors field F finite fixed formula fractional ideal function fundamental given gives global field Hence holds idele identity induces integral invariants isometry isomorphism isotropic Jordan splitting linear matrix maximal multiplication natural non-zero norm obtain orthogonal particular polynomial prime Proof Proposition prove quadratic forms quadratic space rational regular quadratic space represents residue class field respect result ring root rotation satisfies scalar scaling set of spots shows Similarly splitting spots on F square step subgroup Suppose symbol symmetric matrix symmetry Take Theorem theory topological typical unique unit valuation vector vector space write