Let $L/K$ be a finite field extension. Is it true that we can embed $G_L$ into $G_K$?
I think the most natural choice would be to choose the restriction map, so
$$ G_L \to G_K, \quad \sigma \mapsto \sigma|_{\bar{K}}.$$
But I still do struggle to show whether this map is injective or not. Assume we have two different elements $\sigma_1, \sigma_2 \in G_L$. Then there exists an $x \in \bar{L}\setminus L$ with $\sigma_1(x) \neq \sigma_2(x)$. But to show injectivity, I have to find such an element $x$ in $\bar{K}$. Can I rescue this argument?
Thank you!