Evaluation of non-commutative polynomials on a quaternion algebra

יום ד', 13/11/2019 - 10:30
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Let p be a multilinear polynomial in several non-commuting variables, with coefficients in an arbitrary field K.  Kaplansky conjectured that for any n, the image of p evaluated on the set M_n(K) of nxn matrices is a vector space.  We settle the analogous conjecture for a quaternion algebra.