Unramified prime ideal of a number ring

Question

Let $R$ and $S$ be two number ring, $R \subset S$. By definition A prime ideal $P$ is ramified in $S$ if $PS$ is not squarefree in $S$.

Supper I have a number field $F$ and its ring of integers $O$. What does it mean to say that a prime ideal $Q$ of $O$ is unramified? Is it $Q$ is unramifed over $Z$ or $Q$ is unramified in a Galois extension containing F?

Answer 1

As you correctly note, a prime $Q\subset O$ is said to be unramified with respect to some extension $O\subset O'$ of rings of integers if $QO'$ is squarefree. Without giving an extension, the expression 'the prime $Q$ of $O$ is unramified' has no meaning.