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.
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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 Hasse-Minkowski Theorem
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Source Wikipedia: Hasse-Minkowski theorem
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Source Hasse-Minkowski Theorem (Wikipedia)
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1. Wikipedia: Hasse-Minkowski theorem
WikipediaLead 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.
Principal Result Of: Helmut Hasse, Lead sentence of the article on the result
The theorem was proved in the case of the field of rational numbers by Hermann Minkowski and generalized to number fields by Helmut Hasse.
View the Source 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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