  In this paper we provide an algorithm to compute the product between two
elements in a symbol algebra of degree n and we find an octonion non-division
algebra in a symbol algebra of degree three. Starting from this last idea, we
try to find an answer to the question if there are division symbol algebras of
degree three. The answer is positive and we provide, using MAGMA software, some
examples of division symbol algebras of degree 3 and of degree 5. Moreover, we
will give some interesting applications of the symbol algebras in number
theory.
