Prime of a number field (reviewed)

A prime $\mathfrak p$ of a number field $K$ is a nonzero prime ideal of its ring of integers $\mathcal O_K$.

The ideal $\mathfrak p \cap\mathcal O_K$ is a nonzero prime ideal of $\Z$ (a prime of $\Q$), which is necessarily a principal ideal $(p)$ for some prime number $p$. The prime $\mathfrak p$ is then said to be a prime above $p$.