let $G$ be a finite group. let $V$ be an $F$-vector space. and $\rho:G\rightarrow GL(V)$ be a one dimensional representation. I don't see why it is automatically irreducible.
My guess: $V=\langle v_0 \rangle$ and $W$ a proper subspace of $V$ so $W\subset \langle v_0 \rangle$ an element in $W$ is $w=\alpha v_0$ for some $\alpha \in F$ so $\rho (g)(w)=\alpha \rho(g)(v_0)$ but why $W$ can't be invariant by $\rho(g)$ for any $g\in G$.