TITLE: Is ramification of number fields first order?
QUESTION [11 upvotes]: Fix a prime number $p$. Is there a first order sentence $\phi_p$ in the language of fields such that $\phi_p$ holds in a number field $K$ if and only if the prime $p$ is unramified in the field extension $K/\mathbb{Q}$?

REPLY [6 votes]: I THINK, the answer is YES by a paper of Robert Rumely:
Undecidability and definability of the theory of global fields, 1980,
where he showed that "a great variety of number-theoretic objects, from rings of integers and valuations, to zeta-functions and adele rings" are definable. Many of them are actually uniformally definable. 
For the ramification issue, see the bottom of Page 210.