2007/02/15 by A. S. Gadre, Gadre, A. S., S. A. Katre +1
Mathematics · #11R04 #11R11 #11R29 #15A33 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R04 #msc:11R11 #msc:11R29 #msc:15A33
paper · pdf · doi:10.48550/arxiv.math/0702445
9 pages, No figures
arxiv created 2007/02/15 · arxiv updated 2009/12/01
In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We prove a discriminant criterion for every 2 by 2 matrix over an order R to be sums of cubes and fourth powers over R. We also show that if q is a prime and n greater equal 2, then every n by n matrix over the ring of integers O, of a quadratic number field is a sum of q-th powers (of matrices) over O if and only if q is coprime to the discriminant of the quadratic number field.