  Given a number field $K$ and an integer $m\geq 0$, let $K_m$ denote the
maximal $m$-step solvable Galois extension of $K$ and write $G_K^m$ for the
maximal $m$-step solvable Galois group Gal$(K_m/K)$ of $K$. In this paper, we
prove that the isomorphy type of $K$ is determined by the isomorphy type of
$G_K^3$. Further, we prove that $K_m/K$ is determined functorially by
$G_K^{m+3}$ (resp. $G_K^{m+4}$) for $m\geq 2$ (resp. $m \leq 1$). This is a
substantial sharpening of a famous theorem of Neukirch and Uchida. A key step
in our proof is the establishment of the so-called local theory, which in our
context characterises group-theoretically the set of decomposition groups (at
nonarchimedean primes) in $G_K^m$, starting from $G_K^{m+2}$.
