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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
##### Keywords:
trace forms; quadratic forms; field extension
Full Text:
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