论文标题
在数字字段上无限期和潜在的通用二次形式
On indefinite and potentially universal quadratic forms over number fields
论文作者
论文摘要
一个数字字段$ k $允许二进制积分二次形式，该表格代表当地的所有整数，而不是在全球范围内，并且仅当$ k $的班级数大于一个整数时。在这种情况下，只有$ k $以上的二进制积分二次形式只有许多类别的类别。一个数字字段$ k $接收三元积分二次形式，该表格代表当地的所有整数，而不是在全球范围内，并且仅当$ k $的班级数量均匀时。在这种情况下，有很多类别的三元整数二次形式超过$ k $。数字字段$ k $具有多个变量的整体二次形式代表$ k $ $ k $的所有整数（当时$ k $的有限扩展名），并且仅当此二次形式代表$ 1 $ $ k $的整数上的$ 1 $。
A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary integral quadratic forms over $k$. A number field $k$ admits a ternary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is even. In this case, there are infinitely many classes of such ternary integral quadratic forms over $k$. An integral quadratic form over a number field $k$ with more than one variables represents all integers of $k$ over the ring of integers of a finite extension of $k$ if and only if this quadratic form represents $1$ over the ring of integers of a finite extension of $k$.
