Embeddings of quadratic spaces over the field of $p$-adic numbers

Abstract

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 $({\mathbb{Q}}_p^n,x_1^2+\cdots +x_{n-1}^2+\lambda x_n^2)$, where ${\mathbb{Q}}_p$ is the field of $p$-adic numbers, and $\lambda$ is a nonsquare in the finite field ${\mathbb{F}}_p$. Furthermore, the minimum dimension $n$ that admits such an embedding is determined.

14 pages, comments are welcome

Cited by in corpus (1)

• An Isometric Invariant of Quadratic Spaces over Finite Fields