论文标题
非发电的最简单立方场的加性结构
Additive structure of non-monogenic simplest cubic fields
论文作者
论文摘要
我们认为Shanks的最简单的立方字段$ K $为此索引$ [\ Mathcal {o} _K：\ Mathbb {z} [Z} [ρ]] $的root $ρ$的定义参数parametric toluginemial是$ 3 $。对于他们来说，我们研究了$ k $的添加剂的不可分解材料，并提供了完整的列表。此外，我们使用不可分解的知识来证明对$ k $的算术产生一些有趣的后果。主要是，我们在$ k $上获得了通用二次表格等级的良好界限，并证明了$ \ Mathcal {O} _K $的Pythagoras编号为$ 6 $。
We consider Shanks' simplest cubic fields $K$ for which the index $[\mathcal{O}_K:\mathbb{Z}[ρ]]$ of a root $ρ$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of $K$. Mainly, we obtain good bounds on the ranks of universal quadratic forms over $K$ and prove that the Pythagoras number of $\mathcal{O}_K$ is $6$.
