LEVELS AND SUBLEVELS OF QUATERNION ALGEBRAS | Zendy

Detlev W. Hoffmann | Zendy

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LEVELS AND SUBLEVELS OF QUATERNION ALGEBRAS

Author(s) -

Detlev W. Hoffmann

Publication year - 2010

Publication title -

mathematical proceedings of the royal irish academy

Language(s) - English

Resource type - Journals

eISSN - 2009-0021

pISSN - 1393-7197

DOI - 10.3318/pria.2010.110.1.95

Subject(s) - quaternion , mathematics , integer (computer science) , quaternion algebra , division (mathematics) , algebra over a field , division algebra , ring (chemistry) , pure mathematics , division ring , combinatorics , arithmetic , algebra representation , geometry , computer science , chemistry , organic chemistry , programming language

The level s (resp. sublevel s) of a ring R with 1 6= 0 is the smallest positive integer such that −1 (resp. 0) can be written as a sum of s (resp. s+1) nonzero squares in R, provided −1 (resp. 0) is a sum of nonzero squares at all. D.W. Lewis showed that any value of type 2n or 2n +1 can be realized as level of a quaternion division algebra, and in all these examples, the sublevel was 2n, which prompted the question whether or not the level and sublevel of a quaternion division algebra will always differ at most by one. In this note, we give a positive answer to that question.

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