Simultaneous rational approximations to algebraic numbers of QP

In this paper, we prove that if beta(1),...,beta(n) are p-adic numbers belonging to an algebraic number field K of degree n + 1 over Q such that 1, beta(1),...,beta(n) are linearly independent over Z, there exist infinitely many sets of integers (q(0),..., q(n)), with q(0) not equal 0 and \beta(m) - q(m)/q(0)\p much less than H-(1+1/n) log (H)(-1/(n-1)) for m = 1,..., n - 1 \beta(n) - q(n)/q(0)\ much less than H-(1+1/n), with H = H(q(0),..., q(n)). Therefore, these numbers satisfy the p-adic Littlewood conjecture. To obtain this result, we are using, as in the real case by Peck [2], the structure of a group of units of K. The essential argument to obtain the exponent 1 / (n - 1) (the same as in the real case) is the use of the p-adic logarithm. We also prove that with the same hypothesis, the inequalities \beta(m) - q(m)q(0)\(p) less than or equal to epsilonH (q(0),....q(n))(-(1+1/n)) for m = 1,...,n have no integer solution (q(0),...,q(n)) with q(0) not equal 0, if epsilon > 0 is small enough.