四元素代數裡的秩序

Abstract

這篇文章旨在介紹四元素代數裡的秩序。在第二節，我用非常初等的手法，推導出一些我們需要的四元素代數的基本性質；學過一年大學抽象代數並知道一點關於張量積的性質的人應該都能讀的懂。在第三節裡，我證明了秩序的一些性質並引出這篇文章的主角「艾克秩序」。第四節介紹佈於有理數上的秩序的特殊性質。第五節和第六節的目標在描述佈於代數數體上的艾克秩序。在第五節，佈於區域體上的艾克秩序被以清楚明白的式子寫出來。在第六節，我們收集區域的資訊用以瞭解全域體上的艾克秩序。最後，一些有理數上的艾克秩序被清楚地給出。

The goal of this paper is to introduce the quaternion orders. In Section ef{basic}, I give a brief introduction to quaternion algebra. Definitions and basic results necessary for the remaining sections are established in a very elementary way. Anyone who has learned one-year undergraduate algebra and knows the definition of tensor product should be able to read it without difficulties. In Section ef{general}, general properties about the quaternion orders are proven and the Eichler orders are introduced. In Section ef{rational}, special properties of the quaternion orders over the field of rational numbers are given. Section ef{local} and ef{global} are aimed to describe all the Eichler orders over number fields. In Section ef{local}, we give explicit formulations for the Eichler orders over the local field which is the completion of a number field with respect to one of its metric. In Section ef{global}, information about the local Eichler orders are collected to get an understanding about the Eichler orders over global fields. Some Eichler orders over the field of rational numbers are given explicitly.