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Submission: 2006, Aug 2
Herbert Gross asked whether there exist fields admitting anisotropic quadratic spaces of arbitrarily large finite dimensions but none of infinite dimension. This article provides examples of such fields and discusses related problems in the theory of central simple algebras and in Milnor $K$-theory.
2000 Mathematics Subject Classification: 11E04, 11E81, 12E15, 12F20, 12G05, 12G10, 16K20, 19D45
Keywords and Phrases: quadratic form, isotropy, infinite-dimensional quadratic space, $u$-invariant, function field of a quadric, totally indefinite form, real field, division algebra, quaternion algebra, symbol algebra, Galois cohomology, cohomological dimension, Milnor $K$-theory
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