Title: The $θ$-adics Show Chinese title

Title: 该$θ$-adic

Abstract: This paper introduces an archimedean, locally Cantor multi-field $\mathcal{O}_{\theta}$ which gives an analog of the $p$-adic number field at a place at infinity of a real quadratic extension $K$ of $\mathbb{Q}$. This analog is defined using a unit $1<\theta\in \mathcal{O}_{K}^{\times}$, which plays the same role as the prime $p$ does in $\mathbb{Z}_{p}$; the elements of $\mathcal{O}_{\theta}$ are then greedy Laurent series in the base $\theta$. There is a canonical inclusion of the integers $\mathcal{O}_{K}$ with dense image in $\mathcal{O}_{\theta}$ and the operations of sum and product extend to multi-valued operations having at most three values, making $\mathcal{O}_{\theta}$ a multi-field in the sense of Marty. We show that the (geometric) completions of 1-dimensional quasicrystals contained in $\mathcal{O}_{K}$ map canonically to $\mathcal{O}_{\theta}$. The motivation for this work arises in part from a desire to obtain a more arithmetic treatment of a place at infinity by replacing $\mathbb{R}$ with $\mathcal{O}_{\theta}$, with an eye toward obtaining a finer version of class field theory incorporating the ideal arithmetic of quasicrystal rings. Show Chinese abstract

Abstract: 本文介绍了一个阿基米德的局部康托尔多场$\mathcal{O}_{\theta}$，它在实二次扩张$K$的无限点上给出了$p$-进数域的类似物，该扩张是$\mathbb{Q}$的。 这种模拟通过一个单位$1<\theta\in \mathcal{O}_{K}^{\times}$定义，它在$\mathbb{Z}_{p}$中所起的作用与素数$p$相同；$\mathcal{O}_{\theta}$的元素则是以$\theta$为基的贪心Laurent级数。 整数 $\mathcal{O}_{K}$ 有一个规范的包含，其在 $\mathcal{O}_{\theta}$ 中的像稠密，加法和乘法运算可以扩展为最多有三个值的多值运算，使得 $\mathcal{O}_{\theta}$ 在 Marty 的意义上成为一个多重域。 我们证明了包含在 $\mathcal{O}_{K}$ 中的一维准晶体的（几何）完成可以规范地映射到 $\mathcal{O}_{\theta}$。 这项工作的动机部分来自于希望通过将 $\mathbb{R}$ 替换为 $\mathcal{O}_{\theta}$ 来获得对无穷远处的一个更算术的处理，旨在获得一个更精细的类域论版本，结合准晶体环的理想算术。