In 1908, Thue showed that, if
$\zeta$ is a real algebraic number of degree
$n$ and
$\Theta$ is a positive number, the inequality
has only finitely many rational solutions
$p/q$. Siegel, in 1920, showed that one can replace the exponent
$\frac{n}{2}+1+\Theta$ by
\par In this article, Mahler establishes the following
$p$-adic extension of Siegel's result. Let
$f(x)$ be an irreducible polynomial with rational integer coefficients of degree
$n\ge 3$, let
$P_1,P_2,\ldots,P_t$ be finitely many prime numbers and let
$\zeta,\zeta_1,\zeta_2,\ldots,\zeta_t$ be real zero of
$f(x)$, a
$P_1$-adic zero of
$f(x)$, a
$P_2$-adic zero of
$f(x), \ldots $, and a
$P_t$-adic zero of
$f(x)$, respectively. Mahler proves that, if
$\beta$ is Siegel's exponent and
$k\ge 1$ is fixed, the inequality
has finitely many solutions in reduced rational numbers
$p/q$. \par Reprint of the author's paper [Math. Ann. 107, 691--730 (1933; Zbl 0006.10502; JFM 59.0220.01)]. For Part II see [Zbl 1465.11013].
