TITLE: What does it mean for a prime ideal to split completely?
QUESTION [6 upvotes]: See here. What does it mean for a prime ideal to split completely?

REPLY [3 votes]: If $L/K$ is a field extension and $\mathfrak{p}$ a prime of $K$, then in $L$ we have
$$\mathfrak{p}\mathcal{O}_L = \prod_{i=1}^g \mathfrak{P}_i^{e_i}$$
where the $\mathfrak{P}_i$ are primes of $\mathcal{O}_L$ and $e_i\ge 1$ are integers. $\mathfrak{p}$ is totally split if all of the $e_i=1$ and if, further, for each $\mathfrak{P}_i$, the residue field degree
$$f_i = [\mathcal{O}_L/\mathfrak{P}_i : \mathcal{O}_K/\mathfrak{p}] = 1.$$
Note that this means that $g = [L:K]$.