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In this paper we provide an algorithm to compute the product between twoelements in a symbol algebra of degree n and we find an octonion non-divisionalgebra in a symbol algebra of degree three. Starting from this last idea, wetry to find an answer to the question if there are division symbol algebras ofdegree three. The answer is positive and we provide, using MAGMA software, someexamples of division symbol algebras of degree 3 and of degree 5. Moreover, wewill give some interesting applications of the symbol algebras in numbertheory.

Some examples of division symbol algebras of degree 3 and 5