Splitting of prime ideals

Definition

Let $L/K$ be a finite extension of algebraic number fields of degree $n$, with rings of integers $\mathcal{O}_K \subseteq \mathcal{O}_L$. Let $\mathfrak{p}$ be a nonzero prime ideal of $\mathcal{O}_K$. Since $\mathcal{O}_L$ is a Dedekind domain, the ideal $\mathfrak{p}\mathcal{O}_L$ factors uniquely as $$\mathfrak{p}\mathcal{O}_L = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_g^{e_g}$$ with distinct nonzero prime ideals $\mathfrak{P}_i$ of $\mathcal{O}_L$ and integers $e_i \ge 1$. The $\mathfrak{P}_i$ are the primes of $L$ lying over $\mathfrak{p}$. The number $e_i$ is the ramification index of $\mathfrak{P}_i$ over $\mathfrak{p}$, and the residue degree (or inertia degree) is $f_i = [\mathcal{O}_L/\mathfrak{P}_i : \mathcal{O}_K/\mathfrak{p}]$. The prime $\mathfrak{p}$ is said to:

1. split completely in $L$ if $g = n$, that is, $e_i = f_i = 1$ for all $i$;
2. be inert in $L$ if $\mathfrak{p}\mathcal{O}_L$ is a prime ideal, that is, $g = 1$ and $e_1 = 1$;
3. be ramified in $L$ if $e_i > 1$ for some $i$.