Let $G$ be a finite group and $C$ the center of $G$. Let$μ:C→F^×$ be a character of $C$. Prove that there is an irreducible representation $ρ : G → GL(V )$ such that $ρ(c)(v) = μ(c)v$ for all $c ∈ C$ and $v ∈ V$ .
If the field is algebraically closed, then the representation space must be $1-dimensional$. So making elements in $G-C$ act trivially gives an irreducible representation. But for an arbitrary field, I don't even know whether I can apply $<\chi_\rho,\chi_\rho>=1$ iff $\rho$ is irreducible,because it seems that the theorem is based on algebraic closedness of the base field. I know that some other construction can make an extension, like induced representations in wikipedia. But again, I am not sure whether the extension is irreducible.