Scalar Extension of Quadratic Lattices

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<jats:p>Let <jats:italic>E/F</jats:italic> be a finite extension of algebraic number fields, <jats:italic>O<jats:sub>E</jats:sub>, O<jats:sub>F</jats:sub></jats:italic> the maximal orders of <jats:italic>E, F</jats:italic> respectively. A classical theorem of Springer [6] asserts that an anisotropic quadratic space over <jats:italic>F</jats:italic> remains anisotropic over <jats:italic>E</jats:italic> if the degree [<jats:italic>E</jats:italic>: <jats:italic>F</jats:italic>] is odd. From this follows that regular quadratic spaces <jats:italic>U, V</jats:italic> over <jats:italic>F</jats:italic> are isometric if they are isometric over <jats:italic>E</jats:italic> and [<jats:italic>E : F</jats:italic>] is odd. Earnest and Hsia treated similar problems for the spinor genera [2, 3]. We are concerned with the quadratic lattices. Let <jats:italic>L, M</jats:italic> be quadratic lattices over <jats:italic>O<jats:sub>F</jats:sub></jats:italic> in regular quadratic spaces <jats:italic>U, V</jats:italic> over <jats:italic>F</jats:italic> respectively.</jats:p>

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