有理數上的四元素環

Abstract

本論文主要不只是探討四元素環在有理數上的分類情況，也描述由他們所形成的Brauer群結構。 有理數上的四元素環大致可分為2乘2矩陣與可除環情況，而我們可以用quadratic form討論其同構狀況。由於2乘2矩陣均同構，故只需考慮可除環的情況，其可更進一步分為不同的同構狀況。 在局部域的情況，可說明其可除環均同構，並使用希爾伯特符號來分類其為2乘2矩陣或是可除環。 最後我們使用Brauer群來描述其分類，並且闡述其群運算方式，透過Hasse-Minkowski定理我們可以觀察在不同的地方做四元素環局部域的分類，則可以完全決定其在有理數上的分類。

This thesis not only classify all quaternion algebras over rational number field but also describe the group structure of the Brauer group formed by them. The quaternion algebra over rational number field can be roughly classified into two types: the 2 by 2 matrix algebra and division rings. Since all 2 by 2 matrices are isomorphic, we only need to classify division rings into non-isomorphic classes. We study the group of norms and the local Hilbert symbols and show that there are exactly two isomorphic classes of quaternion algebras over the local field unless the field is complex number field. Finally, we classify the quaternion algebras over rational number field and define explicitly the group operation of the Brauer group. By Hasse-Minkowski theorem, a quaternion algebra over the rational number field determines a set of local data and such data determines the quaternion algebra.