A quadratic form over the rational numbers represents zero nontrivially if and only if it does so over the real numbers and over every field of p-adic numbers. It is the archetypal local-global, or Hasse, principle result in number theory.
Facts
StatementTwo quadratic forms over a number field K are equivalent over K if and only if they are equivalent over every completion of K: the real numbers, the complex numbers where relevant, and the p-adic numbers for each prime. Equivalently, a quadratic form over K represents zero nontrivially over K exactly when it does so over every one of these completions. 2 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Proved By
Source Hasse-Minkowski Theorem (Wikipedia)
Sources
1. Wikipedia: Hasse-Minkowski theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
The Hasse-Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e.
2. Hasse-Minkowski Theorem (Wikipedia)
Wikimedia FoundationLead section
The Hasse-Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every topological completion of the field (which may be real, complex, or p-adic).
Proved By: Helmut Hasse, Lead paragraph
The Hasse-Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are
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