I'd like to see a formulation of Dirichlet's theorem for number fields, i.e. some analogue of the assertion:
The number of primes less than $N$ congruent to $a \pmod{m}$ where $(a,m)=1$ is
$\frac{1}{\phi(m)}\pi(N) + o(\pi(N))$
I am really only specifically concerned with such a theorem for the Eisenstein integers $\mathbb{Z}[\omega]$, but I am curious about the general case as well. Is this true, and if so can somebody provide me with a reference? Thanks!
To expand on the excellent comments a bit, one needs both a bit more and a bit less than Chebotarev's density theorem. :-)
Let's take a number field $K$, and a nonzero ideal $\mathfrak{N}$ of the ring $\mathfrak{O}$ of $K$. We divide the nonzero ideals coprime to $\mathfrak{N}$ into a finite number of classes depending on $\mathfrak{N}$. There is an equivalence relation on ideals defined by $\mathfrak{a}\sim\mathfrak{b}$ if $b\mathfrak{a}=a\mathfrak{b}$ where $a$, $b\in\mathfrak{O}$, $a\equiv b\equiv1$ (mod $\mathfrak{N}$) and $a$ and $b$ are totally positive (positive in each embedding of $K$ in $\mathbb{R}$). The equivalence classes are called ray classes modulo $\mathfrak{N}$.
The analogue of Dirichlet's theorem for $K$ is that the prime ideals of $\mathfrak{O}$ are equidistibuted amongst the ray classes. The strong form states that if $\pi_{\mathfrak{a}}(N)$ is the number of prime ideals of norm $\le N$ in the ray class of $\mathfrak{a}$ and $\pi_{\mathfrak{O}}(N)$ is the number of prime ideals of norm $\le N$ in $\mathfrak{O}$ then $$\lim_{N\to\infty}\frac{\pi_{\mathfrak{a}}(N)}{\pi_{\mathfrak{O}}(N)} =\frac1m$$ where $m$ is the number of ray classes modulo $\mathfrak{N}$.
To prove this we need less than Chebotarev's theorem, as we need to apply that only to an abelian extension of $K$, but we need more, namely a suitable extension $L/K$ to apply it to. This extension $L/K$ has abelian Galois group $G$ which is in natural correspondence with the set of ray classes. In detail the Frobenius element attached to a prime ideal $\mathfrak{p}$ is the element of $G$ corresponding to the ray class of $\mathfrak{p}$. This extension exists by the existence theorem of class field theory, quite a deep result.
Let's consider particular examples. For $K=\mathbb{Q}$ all ideals are principal so take $\mathfrak{N}=(N)$ for a positive integer $N$. Then for positive integers $r$ and $s$, the ideals $(r)\sim(s)$ iff there are positive integers $a$ and $b$ congruent to $1$ modulo $N$ with $br=as$. This condition is equivalent to $r\equiv s$ (mod $N$). So ray classes correspond to congruence classes and so we recover Dirichlet's theorem.
Now let $K=\mathbb{Q}(\omega)$. In this case all ideals are principal. Take an ideal $\mathfrak{N}=(\nu)$ of $\mathfrak{O}$. and we find this time that $(\alpha)\sim(\beta)$ if $\alpha\equiv\eta\beta$ (mod $\nu$) where $\eta=\pm \omega^j$ is a unit in $\mathfrak{O}$. (As $K$ has no real embeddings, the condition of total positivity is redundant.) So the analogue of Dirichlet here is that for $\alpha$ coprime to $\nu$ the density of prime ideals $\pi$ with $\pi\equiv\pm\omega^j\alpha$ (mod $\nu$) is indepdendent of the choice of $\alpha$.
