arXiv 2103.053222021-03-09math.NT

复双二次域和CM域中的积分平方和

Sums of integral squares in complex bi-quadratic fields and in CM fields

Srijonee Shabnam Chaudhury

备注：: 11pages. arXiv admin note: text overlap with arXiv:2005.13870

摘要：

设$K$是复双二次域，其环为整数环.对于$K = \mathbb{Q}(\sqrt{-m}$，$\sqrt{n}$），其中$ m \equiv 3 \pmod 4 $和$ n \equiv 1 \pmod 4$，我们证明了每个代数整数都可以写成整数平方和.利用这一点，我们证明了对于任何复双二次域$K$，$4\mathcal{O}_K$的每个元素都可以写成五个整数平方和。此外，我们证明了任何CM域的整数环的毕达哥拉斯数至多为5。此外，我们还给出了两类复双二次域，它们分别为$p(\mathcal{O}_{K})= 3$和$p(4\mathcal{O}_{K})=3 $.这里，$p(\mathcal{O}_{K})$是$K$的整数环的毕达哥拉斯数，而$p(4\mathcal{O}_{K})$是最小的正整数$t$，使得$4\mathcal{O}_{K}$的每个元素都可以写为$t$整数平方和。

英文摘要：

Let $K$ be a complex bi-quadratic field with ring of integers $\mathcal{O}_{K}$. For $K = \mathbb{Q}(\sqrt{-m}$, $\sqrt{n}$), where $ m \equiv 3 \pmod 4 $ and $ n \equiv 1 \pmod 4$, we prove that every algebraic integer can be written as sum of integral squares. Using this, we prove that for any complex bi-quadratic field $K$, every element of $4\mathcal{O}_K$ can be written as sum of five integral squares. In addition, we show that the Pythagoras number of ring of integers of any CM field is at most five. Moreover, we give two classes of complex bi-quadratic fields for which $p(\mathcal{O}_{K})= 3$ and $p(4\mathcal{O}_{K})=3 $ respectively. Here, $p(\mathcal{O}_{K})$ is the Pythagoras number of ring of integers of $K$ and $p(4\mathcal{O}_{K})$ is the smallest positive integer $t$ such that every element of $4\mathcal{O}_{K}$ can be written as sum of $t$ integral squares.

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