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.
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Fields hasn’t non-trivial ideals – janmarqz Jun 14 '18 at 00:41
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But we have https://math.stackexchange.com/questions/1033901/prime-ideals-of-the-ring-of-integers-of-an-algebraic-number-field – janmarqz Jun 14 '18 at 00:44
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2@janmarqz actually it is an abuse of notation that I find everywhere. Many people write prime ideal of $K$ to indicate a prime ideal of $\mathcal O_K$. This is not so confusing thanks to the fact that everybody knows that fields have only trivial ideals. – Blacksmith Jun 14 '18 at 01:26
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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 $K = \mathbb{Q}$.
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1Oh thanks, I knew of course the notion of relative degree, but I didn't think that absolute just means to take $\mathbb Q$ as the base field. – Blacksmith Jun 14 '18 at 01:28
