@MISC{Johansson_adescription, author = {Stefan Johansson}, title = {A Description Of Quaternion Algebras}, year = {} }

Share

OpenURL

Abstract

The main purpose of this paper is to provide an introduction to the arithmetic theory of quaternion algebras. However, it also contains some new results, most notably in Section 5. We will emphasise on the connection between quaternion algebras and quadratic forms. This connection will provide us with an efficient tool to consider arbitrary orders instead of having to restrict to special classes of them. The existing results are mostly restricted to special classes of orders, most notably to so called Eichler orders. The paper is organised as follows. Some notations and background are provided in Section 1, especially on the theory of quadratic forms. Section 2 contains the basic theory of quaternion algebras. Moreover at the end of that section, we give a quite general solution to the problem of representing a quaternion algebra with given discriminant. Such a general description seems to be lacking in the literature. Section 3 gives the basic definitions concerning orders in quaternion algebras. In Section 4, we prove an important correspondence between ternary quadratic forms and quaternion orders. Section 5 deals with orders in quaternion algebras over p-adic fields. The major part is an investigation of the isomorphism classes in the non-dyadic and 2-adic cases. The startingpoint is the correspondence with ternary quadratic forms and known classifications of such forms. From this, we derive representatives of the isomorphism classes of quaternion orders. These new results are complements to existing more ring-theoretic descriptions of orders. In particular, they are useful for computations. Finally, section 6 contains the basic theory of orders in quaternion algebras over algebraic number fields and the connection with the p-adic case. At the end of that section, we give an explicit basis of...