By a “prime” of K (number field), we mean an equivalence class of nontrivial valuations on K. What does it mean for a finite prime p to ramify in an extension L of K? I'm reading these notes http://www.jmilne.org/math/CourseNotes/CFT310.pdf. page 1 he talks about this stuff?
Thank you.
There is a correspondence between nonarchimedean valuations on $K$ and prime ideals of ${\cal O}_K$:
$$v\mapsto \{x\in{\cal O}_K:v(x)>0\}.$$
A finite place ramifies if the corresponding prime ideal ramifies. (A finite place is an equivalence class of nonarchimedean absolute values modulo having the same topology, but such classes of absolute values and valuations are in correspondence as well.) Recall that we say ${\frak p}\triangleleft{\cal O}_K$ ramifies in the extension $L$ if in the prime ideal factorization ${\frak p}{\cal O}_L={\frak P}_1^{e_1}\cdots{\frak P}_g^{e_g}$ at least one of the exponents (called ramification indices) $e_i=e({\frak P}_i|{\frak p})$ is greater than one.
An equivalent condition, that refers to only the valuation information: $v$ ramifies if when extended to $L$ the new value group $v(L^\times)$ contains the original value group $v(K^\times)$ strictly.
I give more detail in this answer.