论文标题
整数在完全虚场的整数环的可确定性和确定性
Definability and decidability for rings of integers in totally imaginary fields
论文作者
论文摘要
我们表明，在$ \ MathBb {Q}^{\ text {tr}} $的整数环中可以在$ \ m \ mathbb {q}^{Q}^{\ text {tr}}（tr}}}（i）的整数中定义。这意味着$ \ mathbb {q}^{\ text {tr}}（i）$的整数环在$ \ mathbb {q}^{\ text {t trext {tr}}}（i）$中是不可确定的，并且不可定义。更一般地，当$ l $是完全真实的field $ k $的完全想象的二次扩展时，我们使用单位组$ r^\ times $的订单$ r \ subseteq \ subseteq \ mathcal {o} _l $来产生存在性可实现的完全定义的完全真实的子集$ x \ subseteq \ subseteq \ subseteq \ nathcal {o {o {o} _ l $。在$ k $的某些条件下，包括$ \ MATHCAL {O} _K $的所谓JR-number是最小值$ \ text {Jr}（\ Mathcal {o} _K）= 4 $，我们推迟了$ \ Mathcal {o} _} _ l $的不确定性。这扩展了以前的工作，在相反的情况下证明了类似的结果$ \ text {jr}（\ Mathcal {o} _K）= \ infty $。特别是，与先前的工作不同，我们不要求$ l $仅包含有限的团结根源。
We show that the ring of integers of $\mathbb{Q}^{\text{tr}}$ is existentially definable in the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$, where $\mathbb{Q}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$ is undecidable and first-order non-definable in $\mathbb{Q}^{\text{tr}}(i)$. More generally, when $L$ is a totally imaginary quadratic extension of a totally real field $K$, we use the unit groups $R^\times$ of orders $R\subseteq \mathcal{O}_L$ to produce existentially definable totally real subsets $X\subseteq \mathcal{O}_L$. Under certain conditions on $K$, including the so-called JR-number of $\mathcal{O}_K$ being the minimal value $\text{JR}(\mathcal{O}_K) = 4$, we deduce the undecidability of $\mathcal{O}_L$. This extends previous work which proved an analogous result in the opposite case $\text{JR}(\mathcal{O}_K) = \infty$. In particular, unlike prior work, we do not require that $L$ contains only finitely many roots of unity.
