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.
$\begingroup$
$\endgroup$
2
-
3$\begingroup$ No. This is in fact equivalent to the inertia degree being $1$. $\endgroup$– WojowuCommented Apr 11, 2022 at 12:31
-
$\begingroup$ Yes, I got it now. Thank you. $\endgroup$– Mumtaj FathimaCommented Apr 11, 2022 at 12:40
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
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 $K=\mathbb{Q}(i)$.
