  Let $K$ be a number field and $d_K$ the absolute value of the discrimant of
$K/\mathbb{Q}$. We consider the root discriminant
$d_L^{\frac{1}{[L:\mathbb{Q}]}}$ of extensions $L/K$. We show that for any
$N>0$ and any positive integer n, the set of length n solvable extensions of
$K$ with root discriminant less than $N$ is finite. The result is motivated by
the study of class field towers.
