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Properties of Quaternion Algebra over the Real Number Field and Z_p

Qin Ying-bing

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Abstract

The ring of quaternion over R,denoted by R[i,j,k],is a quaternion algebra. In this paper,the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed,and the order of its unit group is determined. Lastly,another ring isomorphism of M2(Zp) and the quaternion algebra over Zp when p satisfies some particular conditions are presented.

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What this paper is about

The ring of quaternion over R,denoted by R[i,j,k],is a quaternion algebra. In this paper,the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed,and the order of its unit group is determined. Lastly,another ring isomorphism of M2(Zp) and the quaternion algebra over Zp when p satisfies some particular conditions are presented.

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Available abstract

The ring of quaternion over R,denoted by R[i,j,k],is a quaternion algebra. In this paper,the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed,and the order of its unit group is determined. Lastly,another ring isomorphism of M2(Zp) and the quaternion algebra over Zp when p satisfies some particular conditions are presented.

Key concepts: Quaternion, Quaternion algebra, Mathematics, Isomorphism (crystallography), Ring (chemistry), Algebra over a field, Unit (ring theory), Field (mathematics)

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