Quaternion Algebra over Rational Number Field
Date Issued
2014
Date
2014
Author(s)
Lin, Yun-Xiang
Abstract
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.
Subjects
漢彌爾頓四元素環
希爾伯特符號
Brauer群
Hasse-Minkowski定理
Type
thesis
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ntu-103-R01221021-1.pdf
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