So for a one dimensional Galois representation $\rho: G_{\Bbb Q} \to \mathbb C^{\times}$, I know that it must factor through the abelianization of $G_{\Bbb Q}$, which by the Kronecker-Weber theorem is the Galois group of the maximal cyclotomic extension of $\Bbb Q$. I want to conclude from this that $\rho$ factors through a Dirichlet character, that is, a representation of the Galois group of some finite cyclotomic extension. I've seen this question: Complex Galois Representations are Finite
This definitely gives me the answer, but my question is, can I do this without assuming $\rho$ is continuous? In general how important is the continuity assumption when talking about Galois representations? I'm wondering if this is a purely algebraic fact or if it only applies to continuous representations. I would absolutely accept a reference in lieu of a written answer, surely this is written somewhere but I haven't been able to find it.
As an abstract group, $\Bbb Z_p$ is $q$-divisible for any prime $q \neq p$ and uncountable, while no element is infinitely divisible by $p$, so it is a direct sum of uncountably many copies of $\Bbb Z_{(p)}$ (the localisation of $\Bbb Z$ at $p$).
There are uncountably many group morphisms $\Bbb Z_{(p)} \to \Bbb C^*$ (choose the value at $1$, then for each prime $q \neq p$, you have $q$ choices for the value at $q^{-1}$, again $q$ choices for the value at $q^{-2}$, and so on), so yes, there are many group morphisms $\Bbb Z_p \to \Bbb C^*$, and many group morphisms $\hat {\Bbb Z} \to \Bbb C^*$