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We calculate the topological Hochschild homology groups of a maximal order in a simple algebra over the rationals. Since the positive-dimensional THH groups consist only of torsion, we do this one prime ideal at a time for all the nonzero prime ideals in the center of the maximal order. This allows us to reduce the problem to studying the topological Hochschild homology groups of maximal orders A in simple Q_p-algebras. We show that the topological Hochschild homology of A/(p) splits as the tensor product of its Hochschild homology with THH_*(F_p) We use this result in Brun's spectral sequence to calculate,THH_*(A ; A/(p)), and then we analyze the torsion to get \pi_*(THH(A)^\wedge_p).


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