# Conjugacy Class in Galois Representations

Let $G=Gal(\bar K/K)$ be the absolute Galois group of a number field $K$. Let $v$ be a finite place of $K$ and $w$ a place of $\bar K$ extending $v$. Take an $\ell$-adic representation $$\rho: G \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell}).$$

Assume $\rho$ is unramified at $v$ and denote with $F_{w,\rho}$ the image of the Frobenius element associated at $w$.

How the conjugacy class of $F_{w,\rho}$ does depend only by $v$?

• That is because if you pick a different $w'$ lying above $v$, the Frobenius of $w$ over $v$ and the Frobenius of $w'$ over $v$ are conjugate in $G$. – Ferra Oct 11 '15 at 23:00
• In finite extension I am agree with you. However in this extension I know that you have a little problem because the inertia subgroup it is not trivial. – user262440 Oct 12 '15 at 5:29
• it doesn't matter: the decomposition groups of w and w' are conjugate! – Ferra Oct 12 '15 at 7:51