  Suppose $K$ is a number field and $a_K(m)$ is the number of integral ideals
of norm equal to $m$ in $K$, then for any integer $l$, we asymptotically
evaluate the sum
  \[
  \sum_{m\leqslant T} a_K^l(m)
  \]
  as $T\to\infty$. We also consider the moments of the corresponding Dedekind
zeta function. We prove lower bounds of expected order of magnitude and
slightly improve the known upper bound for the second moment in the non-Galois
case.
