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

TL;DR

This paper classifies which quadratic spaces over $p$-adic fields can be embedded into specific $p$-adic Euclidean and Lorentzian spaces, determining the minimal dimensions for such embeddings using invariants like discriminant and Hasse invariant.

Contribution

It provides a comprehensive classification of embeddable quadratic spaces over $p$-adic fields, including degenerate cases, and identifies the minimal embedding dimensions.

Findings

Characterization of embeddable quadratic spaces based on invariants

Determination of minimal embedding dimensions for Euclidean and Lorentzian $p$-adic spaces

Extension of classification to degenerate quadratic forms

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.