Algebraic number field extensions with prescribed trace form.

*(English)*Zbl 0762.11014Let \(L/F\) 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 \(L/F\).

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 \(L/F\).

Reviewer: H.-J.Bartels (Mannheim)

##### MSC:

11E12 | Quadratic forms over global rings and fields |

11R21 | Other number fields |

12F05 | Algebraic field extensions |

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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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