Abstract
For various types of field extensions [Formula: see text] and values of [Formula: see text] we consider quadratic forms [Formula: see text] that lie in the [Formula: see text]th power [Formula: see text] of the fundamental ideal of [Formula: see text] but are already defined over [Formula: see text]. We then search for some [Formula: see text] of minimal dimension that maps to [Formula: see text] when extending the scalars to [Formula: see text]. This problem can be easily solved completely for finite extensions of odd degree. For quadratic extensions [Formula: see text], the situation is more involved, but solved for [Formula: see text]. For [Formula: see text], we construct quadratic field extensions [Formula: see text] and a form [Formula: see text] such that any form [Formula: see text] that maps to [Formula: see text] when extending the scalars to [Formula: see text] has dimension at least [Formula: see text].
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