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ABSTRACT. An analogue of the convergence part of Khintchine's theorem (1924) for simultaneous approximation of integral polynomials at the pointsIt is a solution of the more general problem than Sprindzuk's problem (1980) in the ring of adeles. We use a new form of the essential and nonessential domain methods in metric theory of Diophantine approximation.

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During the last 10 years the classical Khintchine theorem on approximation of real numbers by rational numbers was generalized for integral polynomials of a fixed degree and for approximation by real algebraic numbers. In this paper we prove a complete analogue of the results for the p-adic case.