Isotropy of Quadratic Spaces in Finite and Infinite Dimension

In the late 1970s, Herbert Gross asked whether there exist fields admitting anisotropic quadratic spaces of arbitrarily large finite dimensions but none of infinite dimension. We construct examples of such fields and also discuss related problems in the theory of central simple algebras and in Milnor $K$-theory.

2000 Mathematics Subject Classification: 11E04, 11E81, 12D15, 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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