Theorem 3.4 of Silverman's Advanced topics in the arithmetic of elliptic curves reads
Theorem 3.4 (Dirichlet's theorem): Let $K$ be a number field and $c$ an integral ideal of $K$. Then every ideal class in $I(c)/P(c)$ contains infinitely many degree $1$ prime of $K$. Here $(I)$ is the group of fractional ideals of $K$ which are relatively prime to $c$ and $P(c)=\{(\alpha):\alpha\in {K^{\times},\alpha\equiv1\bmod c}\}$.
I'll appreciate it if you could give me some reference for the proof of this theorem. Thank you for your help.
What is the meaning of degree $1$ prime, that $\mathfrak{p}\cap \Bbb{Z}$ splits completely in $O_K$?
There are two steps in this theorem, first prove the analytic continuation of $L(s,\psi)=\sum_I N(I)^{-s} \psi(I)$ for each character $\psi:I(c)/P(c)\to \Bbb{C}^\times$, the non-adelic proof is done in Neukirch's ANT, otherwise there is Tate's thesis,
then follow the same arguments as in Dirichlet's theorem on arithmetic progressions, to get that $L(1,\psi)\ne 0$ so that $\prod_\psi L(s,\psi)$ has a pole at $s=1$.