Help with a definition: number theory

Question

I found on some papers in number theory the following: 'Let $K$ be a number field and let $\mathfrak p$ be a prime ideal of $K$ with absolute degree $1$".

What does absolute degree mean? I have ever heard about it. Thank you for your help.

Answer 1

In a context where you're discussing something regarding on a field extension rather than merely a field (e.g. the norm operation), I've seen the adjective "absolute" used to mean that you're taking $\mathbb{Q}$ as the base field.

The degree of a prime ideal (of the ring of integers) is such a notion. I don't have a definition handy so I may have details wrong, but I think it goes like this.

Let $L/K$ be an extension of number fields. Let $\mathfrak{q}$ be a prime ideal of $\mathcal{O}_L$ lying over the prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$. The degree of $\mathfrak{q}$ (over $K$) is the degree of the field extension $[\mathcal{O}_L / \mathfrak{q} : \mathcal{O}_K / \mathfrak{p}]$.

So, the absolute degree would be where you plug in $K = \mathbb{Q}$.

Answer 2

A Google search found this in Encyclopedic Dictionary of Mathematics, Volume 1 (page 219):

a prime ideal of absolute degree $1$ is a prime ideal whose absolute norm is a prime number

The absolute norm is the norm $N_{K/\mathbb Q}$. See also Wikipedia.