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On the scaled trace forms and the transfer of a number field extension. (English) Zbl 0762.11015
We use the same notation as in the previous review. In this paper the following question is mainly examined: For which quadratic forms \(\varphi\) over \(F\) with \(\dim\varphi=mn\) does there exist a form \(\rho\) over \(L\) with \(\dim\rho=n\) such that \(Tr_{L/F}(\rho)=\varphi\)? (Here \(m\) denotes the degree of the field extension \(L\) over \(F\).)
More generally it is shown (Theorem 6): For \(m,n,r\in\mathbb{N}\) and forms \(\varphi_ 1,\dots,\varphi_ r\) over \(F\) with \(\dim\varphi=mn\) there exist an extension \(L/F\) of degree \(m\) and \(\rho_ 1,\dots,\rho_ r\) over \(L\) with \(\dim\rho_ i=n\) such that \(\varphi_ i\cong Tr_{L/F}(\rho_ i)\) for \(i=1,\dots,r\).

MSC:
11E12 Quadratic forms over global rings and fields
11R21 Other number fields
12F05 Algebraic field extensions
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