| Diophantine Approximation is a very important branch of the Number The-ory.In this article,we begin with some fundamental knowledge of the Diophantine Approximation and the diophantine approximation in the p-adic field,after that we considered a theory of the rational approximation in the p-adic field.In 1932,K.Mahler following his fundamental study of the theory of transcendent numbers,formulated the conjecture that for almost all x ∈ R,ω_n(x) = n,where ω_n(x) is defined to be the supremum of the set of real numbers ω for which the inequality|P(ω)| < H(P)~-ωhas infinity many solutions P ∈ P_n .This was proved by Sprinzuk in 1964. In 1969 Sprinzuk considered the p-adic analogue of Mahler's conjecture and obtained Sprinzuk-Mahler theory, that is the inequalityhas at most a finite number of solutions in integer polynomials P of degree n for almost all ω ∈ Q_p. In 1999, V.Bernik and professor yuan jin proved the inhomogeneous case of this theory,they proved that for any d ∈ Q_p the inequalityhas at most a finite number of solutions in integer polynomials P of degree n for almost all ω∈ |
