I'm trying to learn field theory on my own, and one of the papers I am reading has the following sentence:
Now fix an imaginary quadratic field $K=\mathbb{Q}(-d)$, and choose $N$ with the property that the primes dividing $N$ split in $K$. Then clearly we can find an ideal $n$ with $O_K$/$n$ ≡ $\mathbb{Z}/(N)$.
Where $O_K$ is the ring of integers of $K$. I understand what this statement means but the "clearly" part is not immediately obvious to me. I just need someone to spell our exactly why that's true, to get a better sense of what those quotients actually mean.
It suffices to show this for a prime factor of $N$. Such a $p$ will split into a product of primes $\mathfrak{p}_i$. Just chose one of them then we have $$O_K/\mathfrak{p}_i=\mathbb{Z}/p$$ now just take a product of all the primes dividing $N$.