Unramified Prime and Frobenius Element

Let $K/k$ be a finite Galois extension of number fields and let $\mathfrak p$ be a prime of $k$ that is unramified in $K$ (ramification index $e=1$ in $K$).

For a prime $\mathfrak P|\mathfrak p$ of $K$, the decomposition group $D(\mathfrak P|\mathfrak p)\subset \mathrm{Gal}(K/k)$ maps onto $\mathrm{Gal}(\kappa(\mathfrak P)/\kappa(\mathfrak p))$.

The Frobenius element $\mathrm{Frob}_{\mathfrak P}$ is the unique element acting on the residue field by $x\mapsto x^{|\kappa(\mathfrak p)|}$; its conjugacy class $\mathrm{Frob}_{\mathfrak p}$ is independent of $\mathfrak P$.

In the letter: $\sigma$ denotes such a Frobenius and is the Galois component of $\alpha_p$.