What is $\mathscr{D}_{\mathcal{O}_K/\mathbb{Z}}$ where $K$ is a number field?

46 Views Asked by At

This is just a simple notation question, but I am having trouble unpacking some notation on a standard algebraic number theory problem. This is a homework question, but I am being asked to solve a problem from another university class's problem set, so I am not familiar with the notation that is being used (hence why my notes or my text are unhelpful).

The problem is as follows:

Let $d,d'$ be squarefree integers, let $K=\mathbb{Q}(\sqrt{d},\sqrt{d'})$, and let $\mathcal{O}_K$ be the ring of integers of $K$. Let $a\in \mathbb{Z}$ be a generator of $\mathscr{D}_{\mathcal{O}_K/\mathbb{Z}}$ and suppose that $p$ is a prime factor of $a$. Prove that $p\mid 2dd'$. (Hint: First compute the discriminant of $\mathbb{Z}[\sqrt{d},\sqrt{d'}]$.)

My question is simply: what is $\mathscr{D}_{\mathcal{O}_K/\mathbb{Z}}$? As in, literally what does this notation mean? Note that I am not asking for any help on the problem itself.