Introduction to quadratic forms
Introduction to quadratic forms
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DOI：
10.2307/3612223
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发表时间：
1965-12
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影响因子：
--
通讯作者：
O. O’Meara
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文献类型：
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作者：
O. O’Meara
第一部分：域的算术理论I赋值域的赋值阿基米德赋值非阿基米德赋值完全赋值的推广到有限扩展任意赋值的推广到有限可分扩展离散赋值Dedekind理想理论S的Dedekind公理理想理论扩展域数论域有理全局域局部域全局域第二部分：二次型的抽象理论VI二次型与正交群型矩阵与空间二次空间n(V)的特殊子群n(V)的二次型张量积n(V)的中心简单代数上的Wedderburn定理扩展标量域clifford代数n(V)四元数代数的旋量范数特殊子群n(V)的哈塞代数VI二次型的等价完全阿基米德域有限域局部域全局符号全局域上的平方与范数全局域上的二次型VII希尔伯特互易性局部行为规定性形式存在的法律证明二次互易律第四部分环上二次型的算术理论VIII Dedekind域上二次型的抽象格二次空间中的格IX局部域上二次型的积分理论非并矢域上格的分类并矢域上格的分类不变量的有效确定On的特殊子群X整体域上二次型的积分理论算术域上正交群的初等性质类数的格和旋量格的有限性不定情况下的类和旋量格定格的不可分解分裂有理数上的定非模格
Prerequisites ad Notation Part One: Arithmetic Theory of Fields I Valuated Fields Valuations Archimedean Valuations Non-Archimedean valuations Prolongation of a complete valuation to a finite extension Prolongation of any valuation to a finite separable extension Discrete valuations II Dedekind Theory of Ideals Dedekind axioms for S Ideal theory Extension fields III Fields of Number Theory Rational global fields Local fields Global fields Part Two: Abstract Theory of Quadratic Forms VI Quadratic Forms and the Orthogonal Group Forms, matrices and spaces Quadratic spaces Special subgroups of On(V) V The Algebras of Quadratic Forms Tensor products Wedderburn's theorem on central simple algebras Extending the field of scalars The clifford algebra The spinor norm Special subgroups of On(V) Quaternion algebras The Hasse algebra VI The Equivalence of Quadratic Forms Complete archimedean fields Finite fields Local fields Global notation Squares and norms in global fields Quadratic forms over global fields VII Hilbert's Reciprocity Law Proof of the reciprocity law Existence of forms with prescribed local behavior The quadratic reciprocity law Part Four: Arithmetic Theory of Quadratic Forms over Rings VIII Quadratic Forms over Dedekind Domains Abstract lattices Lattices in quadratic spaces IX Integral Theory of Quadratic Forms over Local Fields Generalities Classification of lattices over non-dyadic fields Classification of Lattices over dyadic fields Effective determination of the invariants Special subgroups of On(V) X Integral Theory of Quadratic Forms over Global Fields Elementary properties of the orthogonal group over arithmetic fields The genus and the spinor genus Finiteness of class number The class and the spinor genus in the indefinite case The indecomposable splitting of a definite lattice Definite unimodular lattices over the rational integers Bibliography Index Bibliography Index