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For any field $\\textbf{k} $ of positive characteristics, set $A=\\textbf{k} \\langle x_1,\\dots,x_s\\rangle$ be a free associative algebra, then any centralizer $\\mathcal{C}(f)$ of nontrivial element $f\\in A\\backslash \\textbf{k}$ is a ring of polynomials on a single variable. 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