Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}_l^\times$, where $l$ is a prime and ${\mathbb Q}_l$ the field of $l$-adic numbers. Let $H(\alpha,\beta)$ be the ${\mathbb Q}_l$ vector space of all continuous maps $\chi:G\to{\mathbb Q}_l$ such that $$ \omega=\left(\begin{array}{cc}\alpha&\chi\\\ \ &\beta\end{array}\right) $$ is a group homomorphism.

My question is: is the space $H(\alpha,\beta)$ one-dimensional?

I can prove this if $\alpha=\beta$ and for the proof one can assume that $\beta=1$. In this case the defining relation for $H(\alpha,\beta)$ is $\chi(xy)=\alpha(x)\chi(y)+\chi(x)$.