On In-forms over field extensions.

For various types of field extensions E / F and values of n ∈ ℕ we consider quadratic forms φ that lie in the n th power I n E of the fundamental ideal of E but are already defined over F. We then search for some ψ ∈ I n F of minimal dimension that maps to φ when extending the scalars to E. This problem can be easily solved completely for finite extensions of odd degree. For quadratic extensions E / F , the situation is more involved, but solved for n ≤ 3. For n > 3 , we construct quadratic field extensions E / F and a form φ ∈ I n E such that any form ψ ∈ I n F that maps to φ when extending the scalars to E has dimension at least dim φ + 2 n − 2 . [ABSTRACT FROM AUTHOR]