= Solution
Let $F/K$ be a finite <Galois extension>, choose a <prime ideal> $\mathfrak q$ above $\mathfrak p$, and write $k_{\mathfrak q}$ and $k_{\mathfrak p}$ for the <residue fields>. The <decomposition group> and <inertia group> are
$$
\boxed{D_{\mathfrak q}=\{\sigma\in\operatorname{Gal}(F/K):\sigma\mathfrak q=\mathfrak q\},\qquad
I_{\mathfrak q}=\ker\bigl(D_{\mathfrak q}\longrightarrow
\operatorname{Gal}(k_{\mathfrak q}/k_{\mathfrak p})\bigr).}
$$
The <Frobenius automorphism> on the <finite field> $k_{\mathfrak q}$ is $x\mapsto x^{N\mathfrak p}$. Its inverse image defines a distinguished coset in $D_{\mathfrak q}/I_{\mathfrak q}$; a Frobenius element is a lift of that coset. At an <unramified> prime, $I_{\mathfrak q}=1$ and the lift is unique. Changing the prime above $\mathfrak p$ conjugates it, so a <Frobenius conjugacy class> is intrinsically attached to $\mathfrak p$. These are arithmetic Frobenius conventions.
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