We investigate several topics on a quadratic form $Phi$ over an algebraic number field including the following three: (A) an equation $,xiPhicdotrxi=Psi$ for another form $Psi$ of a smaller size; (B) classification of $Phi$ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the ``class'' of such a $,xi,$ determines a ``class'' in the orthogonal group of a form $Th$ such that $Phi pproxPsiøplusTh.$ Such was done in [S3] when $,Psi$ is a scalar. We will treat the case of nonscalar $Psi,$ and prove a class number formula and a mass formula, both of new types. In [S5] we classified all genera of $Z$-valued $Phi.$ We generalize this to the case of an arbitrary number field, which is topic (B). Topic (C) concerns some explicit forms of the formulas in (A) when $Phi$ is of size 3 and $Psi$ is a scalar.
展开▼
机译：我们在一个代数数字段上研究关于二次形式$ Phi $的几个主题，包括以下三个：（A）另一形式$ Psi的方程$ ， xi Phi cdot tr xi = Psi $较小的$; （B）在代数整数环上对$ Phi $的分类; （三）三元形式。在（A）中，我们证明了这样的$ ， xi ，$的``类别''确定了形式为$ Th $的正交组中的``类别''，使得$ Phi approx Psiøplus Th。$当$ ， Psi $为标量时，此操作在[S3]中完成。我们将处理非标量$ Psi，$的情况，并证明类数公式和质量公式都是新类型。在[S5]中，我们对$ Z $值的$ Phi。$的所有属进行了分类。我们将其归纳为任意数字字段的情况，这是主题（B）。当$ Phi $的大小为3而$ Psi $为标量时，主题（C）涉及（A）中公式的某些显式形式。
展开▼
