Lengths of cyclic algebras and commutative subalgebras of quaternion matrices

Abstract This paper contains two parts. In the first we determine the length of the central simple cyclic algebras of degree less than or equal to five. In the second we give a upper bound for the length of a commutative subalgebra of the full matrix algebra over the real quaternions. We also prove that if the length of a commutative matrix subalgebra is maximal, then this subalgebra is maximal under inclusion.