There are no universal ternary quadratic forms over biquadratic fields.(English)Zbl 1460.11042
The main result is the following theorem: For any totally real biquadratic number field \(K\) there are no universal classic totally positive definite ternary quadratic forms over the ring of algebraic integers \(O_K\).
The same conclusion is correct for all totally real number fields \(K\) with \(\sqrt{2}\not\in K\) which contain a nonsquare totally positive unit \(\varepsilon\) such that \(2\varepsilon\) is not a square in \(K\). The main tools for the proof are results about additively indecomposable elements of \(O_K\).
The same conclusion is correct for all totally real number fields \(K\) with \(\sqrt{2}\not\in K\) which contain a nonsquare totally positive unit \(\varepsilon\) such that \(2\varepsilon\) is not a square in \(K\). The main tools for the proof are results about additively indecomposable elements of \(O_K\).
Reviewer: Meinhard Peters (Münster)
MSC:
| 11E20 | General ternary and quaternary quadratic forms; forms of more than two variables |
| 11E12 | Quadratic forms over global rings and fields |
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11R80 | Totally real fields |
Keywords:
indecomposable integer;universal quadratic form;ternary quadratic form;biquadratic number fieldCite
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