论文标题
$ p $ - adic数字的二次空间的嵌入
Embeddings of quadratic spaces over the field of $p$-adic numbers
论文作者
论文摘要
$ p $ adiC字段上的非二次二次形式由其尺寸，判别和哈斯不变。本文使用这三个不变性，关于$ p $ - adic领域的基本事实和二次形式的理论来确定哪些类型的二次空间（包括退化案例）可以嵌入欧几里得$ p $ - 亚种空间中$） $（\ mathbb {q} _ {p}^{n}，x_ {1}^{2}+\ cdots+x_ {n-1}^{2}+λx_{n}^n}^{2}^{2}^{2}）$在有限字段$ \ mathbb {f} _ {p} $中的非Quare。此外，确定了承认这种嵌入的最小尺寸$ n $。
Nondegenerate quadratic forms over $p$-adic fields are classified by their dimension, discriminant, and Hasse invariant. This paper uses these three invariants, elementary facts about $p$-adic fields and the theory of quadratic forms to determine which types of quadratic spaces -- including degenerate cases -- can be embedded in the Euclidean $p$-adic space $(\mathbb{Q}_{p}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$, and the Lorentzian space $(\mathbb{Q}_{p}^{n},x_{1}^{2}+\cdots+x_{n-1}^{2}+λx_{n}^{2})$, where $\mathbb{Q}_{p}$ is the field of $p$-adic numbers, and $λ$ is a nonsquare in the finite field $\mathbb{F}_{p}$. Furthermore, the minimum dimension $n$ that admits such an embedding is determined.
