I encountered this in this paper:
Suppose $\rho$ factors through a finite cyclic group $G = \operatorname{Gal}(F/K) \simeq C_n$.
I know a little bit about representation theory, but unfortunately, I have never heard of factoring through a group before, so I do not know what exactly this is supposed to mean. Could you please explain that to me? Thank you!
On
The reason for this "factoring" terminology is that homomorphisms between groups behave, to a certain extent, like multiplication: there is a binary operation called "composition"; and this operation satisfies the associative law. For example, we teach our students to factor functions as compositions in order to apply the chain rule: $f(x)=\cos(x^2)$ factors as $f=g \circ h$ where $h(x)=x^2$ and $g(x)=\cos(x)$.
However, one way this differs from multiplication (besides the failure of commutativity) is that the domains and ranges need not all be the same. The point of "factoring through $G$" is that one can factor the representation so that the group in the middle is $G$. Denoting your given group representation as $\rho : K \to \Gamma$ (where $\Gamma$ might be the general linear group of some vector space, or a Lie group, or any other kind of group), this means that there are two group homomorphisms $$K \xrightarrow{\sigma} G \xrightarrow{\tau} \Gamma $$ such that $\rho = \tau \circ \sigma$.
Generally speaking, you can see (or even define) a representation $\rho$ of a group $G$ as a group homomorphism $G\to GL(V)$ for some vector space $V$. Then saying that the representation factors through a group $H$ means that this group homomorphism factors through $H$, in the sense that there exist group homomorphisms $\varphi: G\to H$ and $\psi: H \to GL(V)$ such that $\rho=\psi\circ \varphi$.
If you prefer to see a representation as a certain function $G\times V\to V:(g,v) \mapsto g\ast_\rho v$ satisfying certain conditions, then the above tells you that $g\ast_\rho v=\varphi (g)\ast_\psi v$ for all $g,v$.