## Reprint: Zur Approximation algebraischer Zahlen. I: Über den größten Primteiler binärer Formen (1933)

##### Doc. Math. Extra Vol. Mahler Selecta, 333-366 (2019)
DOI: 10.25537/dm.2019.SB-333-366

### Summary

In 1908, Thue showed that, if $\zeta$ is a real algebraic number of degree $n$ and $\Theta$ is a positive number, the inequality $\left|\frac{p}{q}-\zeta\right|\le q^{-\left(\frac{n}{2}+1+\Theta\right)}$ has only finitely many rational solutions $p/q$. Siegel, in 1920, showed that one can replace the exponent $\frac{n}{2}+1+\Theta$ by $\beta=\min_{1\le s\le n-1}\left(\frac{n}{s+1}+s+\Theta\right).$ \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 $\min\left\{1,\left|\tfrac{p}{q}-\zeta\right|\right\} \prod_{\tau=1}^t \min\{1,|p-q\zeta_\tau|_{P_\tau}\}\le k\, \max\{|p|,|q|\}^{-\beta}$ 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].

11-03, 11J68