Irrationality of Dedekind zeta values For Riemann's zeta function, one knows that:
$\zeta(2n)$ is irrational (because a rational multiple of $\pi^{2n}$ is)
$\zeta(3)$ is irrational (proved by Apéry)
and a few other results like "there are infinitely many irrational values at odd integers" (Ball-Rivoal).
Is there something known for the values of the Dedekind's zeta function $\zeta_K(s)$ of a number field? For instance, do we know the irrationality of $\zeta_K(2)$ in some cases? 
 A: As with all motivic L-functions, irrationality of special values breaks into two rather different problems:
Critical values. In the case of Dedekind zeta functions, those are the even positive integers, and the strategy is to show that $\zeta_K(-2n-1)$ is rational, so that from the functional equation follows that $\zeta_K(2n)$ is irrational. Of course, this only works for totally real number fields, since that's the only case where the functional equation doesn't vanish at the odd integers.
This is precisely the Siegel-Klingen theorem, $\zeta_K(2n)$ is a rational multiple of $\pi^{2n[K:\mathbb{Q}]}$, and therefore irrational.
The actual value of that algebraic number is in general not known, see the Lichtenbaum conjectures.
Non-critical values. Odd positive integers. This is the (even more) difficult part, because you have to actually prove that something non-trivial is irrational. As you mention, some results are known for Riemann's zeta, but as far as I know, there are no known results for number fields other than $\mathbb{Q}$.
(On a side note, a generalized "infinitely many irrational numbers" type result for Dirichlet L-functions was published a few years ago by Masaki Nishimoto (paper here)).
