# Integers in residue classes $\mathcal{O}_K/\mathfrak{p}$

Let $$K$$ be a number field and $$\mathcal{O}_K$$ be its ring of integers. For any prime ideal $$\mathfrak{p}$$ in $$\mathcal{O}_K$$ is it true that every residue class in $$\mathcal{O}_K/\mathfrak{p}$$ contains an integer? I can prove that it is true if $$\mathfrak{p}$$ is unramified and has inertial degree 1, but not for general prime ideals. I kindly request your answers to this problem.

• No. This is in fact equivalent to the inertia degree being $1$. Commented Apr 11, 2022 at 12:31
• Yes, I got it now. Thank you. Commented Apr 11, 2022 at 12:40

Let $$p$$ be the prime number that satisfies $$p\mathbb{Z} = \mathbb{Z} \cap \mathfrak{p}$$. Then your claim is equivalent to the inclusion $$\mathbb{F}_p\subset \mathcal{O}_K/\mathfrak{p}$$ being an equality, in other words the prime $$\mathfrak{p}$$ having inertial degree $$1$$. So it is not true in general, for example it is not true for the prime ideal $$3\mathcal{O}_K$$ if $$K=\mathbb{Q}(i)$$.