We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

中文翻译：

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双二次域上没有普遍的三元二次形式

我们研究整数环上的完全正定二次型$\数学{O}_K$一个完全真实的双二次域$K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. 我们将注意力限制在经典形式上（即那些在$2\数学{O}_K$）并证明三个变量中没有这样的形式是普遍的（即代表所有完全积极的元素$\数学{O}_K$）。此外，对于包含至少一个非平方全正单元并满足其他一些温和条件的全实数域，我们展示了相同的结果。这些结果为北冈的猜想提供了进一步的证据，即只有有限多个数域存在这种形式。我们的主要工具之一是添加不可分解的元素$\数学{O}_K$; 我们证明了一些关于它们属性的新结果。