Can prime (Ideals) be ramified/split completely in 'their own field'? Recently I have come across a few sources where the definitions of primes being ramified or splitting completely do not quite adhere to the way I learned them. I completely understand the 'standard' definitions, where we take $L/K$ to be a finite (algebraic number) field extension, and then choose $\mathfrak{p}\in \mathcal{O}_K$, and then determine how $\mathfrak{p}\mathcal{O}_L$ splits up. (ie whether any of the $\mathfrak{P}_i\in\mathcal{O}_L$ containing $\mathfrak{p}$ have a power greater than one in the unique factorisation.) 
My question is this: Can we give definitions of ramified, split, inert, without defining a 'larger' field? I ask because in two separate sources I have seen (http://arxiv.org/pdf/1108.5671.pdf, lemma 2.2 and Washington: Introduction to Cyclotomic Fields, Chapter 15) The author seems to state that a prime $\mathfrak{p}$ splits completely in $K$, where $\mathfrak{p}$ is in $K$. 
I don't quite understand how we define this.
I am sure that there is some simple terminology or obvious thing that I am missing but I can't seem to work it out at the minute.
Any help would be greatly appreciated. 

For primes of a number field $F$, you can always talk about ramification, inertia, etc of the underlying prime in the finite extension $F|\mathbb{Q}$. For example, the article you are referring to states:

Let $\mathfrak q$ be a prime ideal in $F$ [...] then $\mathfrak q$ splits completely in $F|\mathbb{Q}$.

So if $\mathfrak q$ lies over $q\in \mathbb{Z}$, then $q$ is completely split in the extension $F|\mathbb{Q}$.
