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Algebraic number field extensions with prescribed trace form. (English) Zbl 0762.11014
Let $$L/F$$ be a finite extension of algebraic number fields. A quadratic form over $$F$$ is called positive, if the signature of this form is nonnegative for all orderings of $$F$$. The trace form $$Tr_{L/F}\langle 1\rangle$$ is a positive form for example. In the paper under review the following question of Conner and Perlis is investigated: Which positive quadratic forms over $$F$$ are isometric to a trace form of some field extension $$L/F$$? Definition: $$(\text{tr }n)$$ holds, if for every number field $$F$$ every positive form $$\varphi$$ over $$F$$ with $$\dim\varphi=n$$ is isometric to a trace form of some field extension $$L/F$$ with $$[L:F]=n$$.
The main result of the paper is the following Theorem: Let $$n\geq 4$$. Then (i) if $$(\text{tr }n)$$ holds then $$(\text{tr} nm)$$ holds for all $$m\in\mathbb{N}$$, (ii) $$(\text{tr} n)$$ holds if $$n$$ is divisible by 2 or 3.
Remark: The question in the case of quadratic forms in dimension 2, 3 and 4 was already answered by Conner and Perlis. For the proof some number theory is needed, for instance the approximation theorem, Dirichlet’s density theorem and the ramification of prime ideals in $$L/F$$.

##### MSC:
 11E12 Quadratic forms over global rings and fields 11R21 Other number fields 12F05 Algebraic field extensions
##### Keywords:
trace form; positive quadratic forms; field extension
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##### References:
 [1] Conner, P.E.; Perlis, R., () [2] Endler, O., () [3] Estes, D.; Hurrelbrink, J.; Perlis, R., Total positivity and algebraic Witt classes, Comment. math. helv., 60, 284-290, (1985) · Zbl 0589.10021 [4] Krüskemper, M., Algebraic systems of quadratic forms of number fields and function fields, Man. math., 65, No. 2, 225-243, (1989) · Zbl 0689.10027 [5] Krüskemper, M., On the scaled trace forms and the transfer of a number field extension, J. number theory, 40, 105-119, (1992) · Zbl 0762.11015 [6] Scharlau, W., () [7] Scharlau, W., On trace forms of algebraic number fields, Math. Z., 196, 125-127, (1987) · Zbl 0658.10025
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