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.